You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
2024W-Prob1__script.pdf: sections 1–5 (up to and including 5. Lebesgue-Integral)
2025W-Prob1-Collection.pdf: sections 0–3 (up to and including 3. Das Lebesgue-Integral)
Notes:
Warning: I dropped \mathcal, since it doesn't render in gist
Likewise, _\# became _H
The items have rough importance rankings
roughly "important" vs "neutral" and "random"; I'll derank a few later
The exercises have a "done" if we actually did them in the session
[Script 2024] 1. Messbare Räume und Maße
Definitionen
Definition 1.1: Algebra $\star\star\star$
A family of subsets of $\Omega$ that contains $\Omega$ and is closed under complements and finite unions (equivalently finite intersections).
Definition 1.2: σ-Algebra $\star\star\star$
A family of subsets of $\Omega$ that contains $\Omega$ and is closed under complements and countable unions (equivalently countable intersections).
Definition 1.3: Messbarer Raum $\star\star\star$
A measurable space is a pair $(\Omega,{A})$ where ${A}$ is a σ-algebra on $\Omega$; sets in ${A}$ are called measurable.
Definition 1.5: Maß $\star\star\star$
A measure on $(\Omega,{A})$ is a map $\mu:{A}\to[0,\infty]$ with $\mu(\emptyset)=0$ and countable additivity on pairwise disjoint measurable sets.
σ-finite, finite measures, and probability measures are defined via additional total mass conditions.
Definition 1.6: Maßraum / Wahrscheinlichkeitsraum $\star\star\star$
A measure space is a triple $(\Omega,{A},\mu)$.
If $\mu$ is a probability measure $P$, the triple is called a probability space.
Definition 1.10: Atom eines Maßes $\star\star$
A point $\omega\in\Omega$ is an atom if ${\omega}$ is measurable and has positive measure under $\mu$.
Definition 1.13: Von $M$ erzeugte σ-Algebra $\sigma(M)$$\star\star\star$
For $M\subseteq{P}(\Omega)$, $\sigma(M)$ is the intersection of all σ-algebras containing $M$, hence the smallest σ-algebra that contains $M$.
Definition (Spur / Trace) $\star\star$
For $K\subseteq\Omega$ and a family $M\subseteq{P}(\Omega)$, the trace of $M$ on $K$ is $M\upharpoonright K := {M\cap K : M\in M}$, i.e. all intersections of sets in $M$ with $K$.
Theoreme
Satz 1.7: Eigenschaften von Maßen $\star\star\star$
Collects basic properties of measures: finite additivity on finitely many disjoint sets, monotonicity, σ-subadditivity, and the inclusion–exclusion formula for two sets.
Korollar 1.8: Maß von Differenzmengen und Komplementen $\star\star$
For $A\subseteq B$ with finite $\mu(B)$ one has $\mu(B\setminus A)=\mu(B)-\mu(A)$.
For finite measures, $\mu(A^c)=\mu(\Omega)-\mu(A)$ (for probabilities $ P(A^c)=1-P(A)$).
Satz 1.9: Stetigkeit von unten und oben $\star\star\star$
Measures are continuous from below for increasing sequences of sets and from above for decreasing sequences with finite starting measure.
Proposition 1.11: Abzählbarkeit der Atome in σ-endlichen Räumen $\star$
In a σ-finite measure space, the set of atoms is at most countable.
Lemma 1.12: Durchschnitt von σ-Algebren $\star\star$
The intersection of any family of σ-algebras on the same base set is again a σ-algebra.
Lemma 1.14: Monotonie der erzeugten σ-Algebra $\star\star$
If $M_1\subseteq M_2\subseteq{P}(\Omega)$, then $\sigma(M_1)\subseteq\sigma(M_2)$.
Proposition 1.15: Spur-σ-Algebra $\star\star$
For a σ-algebra ${A}$ on $\Omega$ and $K\subseteq\Omega$, the trace ${A}\upharpoonright K$ is a σ-algebra on $K$, called the trace σ-algebra.
Lemma 1.16: Spur und Erzeugung vertauschen $\star\star$
For $M\subseteq{P}(\Omega)$ and $K\subseteq\Omega$ one has $\sigma(M\upharpoonright K)=\sigma(M)\upharpoonright K$, i.e. generating a σ-algebra and taking traces commute.
[Script 2024] 2. Äußere und Innere Maße
Definitionen
Definition 2.1: λ-System (Dynkin-System) $\star\star$
A family $D\subseteq{P}(\Omega)$ is a λ-system if it contains $\Omega$, is closed under taking differences of nested sets, and under countable increasing unions.
Definition 2.3: π-System $\star\star$
A family $M\subseteq{P}(\Omega)$ is a π-system if it is closed under finite intersections.
Definition 2.5: Von $M$ erzeugtes λ-System $\lambda(M)$$\star\star$ $\lambda(M)$ is the intersection of all λ-systems containing $M$, hence the smallest λ-system containing $M$.
Definition 2.10: Prämaß $\star\star\star$
A premeasure on an algebra ${A}_0$ is a map $\mu:{A}_0\to[0,\infty]$ which is countably additive on disjoint sequences whose union stays in ${A}_0$.
Definition 2.11: Äußeres Maß $\mu^*$$\star\star\star$
Given a premeasure $\mu$ on ${A}_0$, the outer measure $\mu^*:{P}(\Omega)\to[0,\infty]$
is defined by covering sets with countable unions of ${A}_0$-sets and taking an infimum of sums of $\mu$.
Definition 2.12: Von außen messbare Mengen, ${A}^*$$\star\star$
A set $A\subseteq\Omega$ is outer measurable (w.r.t. $\mu^*$) if $\mu^*(M\cap A)+\mu^*(M\cap A^c)=\mu^*(M)$
for all $M\subseteq\Omega$; ${A}^*$ is the family of all such sets.
Theoreme
Lemma 2.2: Gleichheitsmenge zweier Maße ist λ-System $\star\star$
For two finite measures $\mu,\nu$ on $(\Omega,{A})$ with equal total mass, the set $D={A\in{A}:\mu(A)=\nu(A)}$
is a λ-system.
Lemma 2.4: λ- und π-System ⇒ σ-Algebra $\star\star$
If $M$ is both a λ-system and a π-system, then $M$ is a σ-algebra.
Proposition 2.6: Minimalität von $\lambda(M)$$\star\star$ $\lambda(M)$ is itself a λ-system and the smallest λ-system containing $M$.
Satz 2.7 (Sierpiński–Dynkin λ–π Theorem) $\star\star\star$
If $M$ is a π-system, then $\lambda(M)=\sigma(M)$, i.e. the λ-system generated by $M$ coincides with the σ-algebra generated by $M$.
Korollar 2.8: Eindeutigkeit endlicher Maße auf $\sigma(M)$$\star\star\star$
If two finite measures agree on a π-system $M$ and have the same total mass, then they agree on the σ-algebra $\sigma(M)$.
Korollar 2.9: Eindeutigkeit bei σ-Endlichkeit $\star\star$
If two measures agree on a π-system $M$ and one is σ-finite on $M$, then the measures agree on $\sigma(M)$.
Lemma 2.13: Grundlegende Eigenschaften von $\mu^*$$\star\star$
Outer measure $\mu^*$ is zero on the empty set, nonnegative, monotone, and σ-subadditive.
Korollar 2.14: Endliche Subadditivität und Charakterisierung der Außenmessbarkeit $\star\star$
Outer measures are finitely subadditive, and outer measurability can be checked with a $\le$ inequality rather than equality.
Lemma 2.15: ${A}^*$ ist eine Algebra $\star\star$
The class of outer measurable sets ${A}^*$ is closed under complements and finite unions, hence forms an algebra.
Lemma 2.16: σ-Additivität von $\mu^*$ auf disjunkten Familien in ${A}^*$$\star\star$
For disjoint outer measurable sets $A_i$ one has $\mu^*(M\cap\bigcup_i A_i)=\sum_i\mu^*(M\cap A_i)$ for all $M$.
Lemma 2.17: ${A}^*$ ist eine σ-Algebra $\star\star$
The class ${A}^*$ is closed under countable unions, hence is a σ-algebra.
Lemma 2.18: Übereinstimmung von $\mu$ und $\mu^*$ auf ${A}_0$$\star\star$
For every $A\in{A}_0$, the outer measure $\mu^*(A)$ coincides with the original premeasure $\mu(A)$.
Lemma 2.19: ${A}_0 \subseteq {A}^*$$\star\star$
Every set in the original algebra ${A}_0$ is outer measurable, so ${A}_0\subseteq{A}^*$.
Satz 2.20: Aus $(\mu,{A}_0)$ entsteht ein Maßraum $(\Omega,{A}^*,\mu^*)$$\star\star\star$
The outer measurable sets ${A}^*$ form a σ-algebra and $\mu^*$ is a measure on $(\Omega,{A}^*)$ extending the premeasure $\mu$ and agreeing with it on ${A}_0$.
Satz 2.21 (Maßerweiterungssatz von Carathéodory) $\star\star\star$
A σ-finite premeasure on an algebra admits a unique extension to a measure on the σ-algebra generated by the algebra.
[Script 2024] 3. Maß auf $\mathbb{R}$
Definitionen
Definition 3.3: Halboffene Intervalle, Familie $J$ und $\varphi$$\star\star$
For $-\infty\le a\le b\le\infty$, the half-open interval $(a,b]$ and family $J$ of such intervals are defined, and $\varphi((a,b]) = F(b)-F(a)$ (or $0$ if $a=b$) for a given monotone right-continuous $F$.
Definition 3.5: Algebra $J^*$$\star\star$ $J^*$ consists of all finite disjoint unions of half-open intervals from $J$, forming an algebra on $\mathbb{R}$.
Definition 3.7: Erweiterung $\varphi^*$ auf $J^*$$\star\star$
For $A\in J^*$ written as a finite disjoint union of $(a_i,b_i]$, define $\varphi^*(A):=\sum_i \varphi((a_i,b_i])$, extending $\varphi$ from $J$ to $J^*$.
Definition 3.13: Borelsche σ-Algebra ${B}(\mathbb{R})$$\star\star\star$
The Borel σ-algebra on $\mathbb{R}$ is defined as ${B}(\mathbb{R}) := \sigma(J^*)$, the σ-algebra generated by $J^*$ (equivalently by various interval families).
Definition 3.18: Lebesgue-Maß $\lambda$$\star\star\star$
Lebesgue measure $\lambda$ on $\mathbb{R}$ is the measure induced by $F(x)=x$ via the construction of $\mu_F$.
Definition 3.19: Verteilungsfunktion (cdf) $\star\star\star$
A distribution function is a monotone non-decreasing, right-continuous $F:\mathbb{R}\to[0,1]$ with limits $0$ at $-\infty$ and $1$ at $+\infty$, inducing a probability measure on $\mathbb{R}$.
Theoreme
Beispiel 3.1: Diskrete Verteilungen auf $\mathbb{N}$$\star\star$
Constructs a probability measure on $(\mathbb{N},{P}(\mathbb{N}))$ from a sequence of nonnegative weights summing to $1$, covering all discrete distributions on $\mathbb{N}$.
Satz 3.2: Umordnung absolut konvergenter Reihen $\star\star$
Absolutely convergent series can be reordered arbitrarily without changing their sum.
Satz (Riemann’scher Umordnungssatz) $\star$
If a series is conditionally but not absolutely convergent, then for any $a\in\mathbb{R}$ there is a rearrangement whose sum is $a$.
Lemma 3.4: σ-Additivität von $\varphi$ auf $J$$\star\star$
Shows that $\varphi$ is countably additive on $J$ whenever disjoint half-open intervals have a union still in $J$.
Lemma 3.6: $J^*$ ist eine Algebra $\star\star$
Shows that $J^*$ is closed under complements and finite intersections, hence is an algebra.
Proposition 3.8: Wohldefiniertheit von $\varphi^*$$\star\star$
Proves that $\varphi^*$ does not depend on the particular representation of a set in $J^*$ as a disjoint union of basic intervals.
Lemma 3.9: Eigenschaften von $\varphi^*$$\star\star$
Establishes that $\varphi^*$ is a premeasure on $J^*$: zero on $\emptyset$, σ-additive on disjoint unions in $J^*$, and σ-finite.
Lemma 3.10: σ-Additivität auf $J$ vs. $J^*$$\star\star$
States that σ-additivity of $\varphi$ on $J$ is equivalent to σ-additivity of $\varphi^*$ on $J^*$.
Lemma 3.11: Untere Abschätzung durch Zerlegung $\star\star$
If disjoint intervals $(a_i,b_i]$ lie inside $(a,b]$, then $\sum_i(F(b_i)-F(a_i))\le F(b)-F(a)$.
Lemma 3.12: Obere Abschätzung durch Überdeckung $\star\star$
If $(a,b]$ is contained in a union of intervals $(a_i,b_i]$, then $F(b)-F(a)\le \sum_i(F(b_i)-F(a_i))$.
Proposition 3.14: Alternative Erzeuger von ${B}(\mathbb{R})$$\star\star$
Shows that the Borel σ-algebra can equally be generated by open intervals, closed intervals, rays of the form $(-\infty,b]$ or $(-\infty,b)$.
Proposition 3.15: ${B}(\mathbb{R}) = \sigma(\text{offene Mengen})$$\star\star$
Shows that ${B}(\mathbb{R})$ is the σ-algebra generated by all open subsets of $\mathbb{R}$.
Korollar 3.16: Einpunkt-, offene und abgeschlossene Mengen sind borelsch $\star\star$
Concludes that singletons, open sets, and closed sets in $\mathbb{R}$ are all Borel measurable.
Satz 3.17: Existenz und Eindeutigkeit von $\mu_F$$\star\star\star$
For each monotone right-continuous $F$, there exists a unique σ-finite measure $\mu_F$ with $\mu_F((a,b])=F(b)-F(a)$.
Satz 3.20: Maß ⇒ Verteilungsfunktion $\star\star\star$
Conversely, any σ-finite measure on $(\mathbb{R},{B}(\mathbb{R}))$ arises from a monotone right-continuous $F$ via $\varphi((a,b])=F(b)-F(a)$.
Korollar 3.21: Atome und Sprungstellen $\star\star$
For such measures, $\mu({a})$ equals the jump of $F$ at $a$, so atoms correspond exactly to jump points of $F$.
[Script 2024] 4. Messbare Abbildungen und Zufallsvariablen
Definitionen
Definition 4.1: Urbild unter $f$$\star\star$
For $f:\Omega\to\Omega'$, the preimage $f^{-1}(A')={\omega\in\Omega : f(\omega)\in A'}$
is defined for all $A'\subseteq\Omega'$.
Definition 4.2: Messbare Abbildung und Zufallsvariable $\star\star\star$
A map $f:(\Omega,{A})\to(\Omega',{A'})$ is ${A}$–${A'}$-measurable if $f^{-1}(A')\in{A}$ for all $A'\in{A'}$.
Such an $X:(\Omega,{A},P)\to(\Omega',{A'})$ is called an $\Omega'$-valued random variable.
Definition 4.3: Von $(f_i)$ erzeugte σ-Algebra $\sigma(f_i, i\in I)$$\star\star$
For maps $f_i:\Omega\to\Omega'$, the σ-algebra they generate is $\sigma(f_i, i\in I):=\sigma({f_i^{-1}(A') : A'\in{A'}, i\in I})$.
Definition 4.10: Erweiterte reelle Zahlen und ${B}(\overline{\mathbb{R}})$$\star\star$
The extended reals $\overline{\mathbb{R}}=\mathbb{R}\cup{-\infty,\infty}$ are introduced, with $K={[-\infty,t] : t\in\mathbb{R}}$ and ${B}(\overline{\mathbb{R}}):=\sigma(K)$ as the Borel σ-algebra on $\overline{\mathbb{R}}$.
Definition 4.11: Rechenregeln in $\overline{\mathbb{R}}$$\star\star$
Specifies how to add and multiply finite real numbers with $\pm\infty$ (and that $\infty-\infty$ is undefined) to make sense of arithmetic in $\overline{\mathbb{R}}$.
Definition 4.16: Indikatorfunktion $1_A$$\star\star$
The indicator function $1_A:\Omega\to{0,1}$ takes value $1$ on $A$ and $0$ outside $A$.
Definition 4.19: Einfache Funktion $\star\star\star$
A simple function is a finite linear combination $f=\sum_{i=1}^n \alpha_i 1_{A_i}$
with measurable sets $A_i$ and real coefficients $\alpha_i$.
Theoreme
Proposition (nach 4.1): Urbildoperator und Mengenoperationen $\star\star$
Shows that preimages commute with unions, intersections, and complements (and hence with countable unions and intersections).
Proposition 4.4: Messbarkeit über einem Erzeuger $M'$$\star\star$
If ${A'}=\sigma(M')$ and $M={f^{-1}(M') : M'\in M'}$, then $\sigma(f)=\sigma(M)$ and $f$ is measurable iff $M\subseteq{A}$.
Lemma 4.5: Alternative Beschreibung von $\sigma(f_i)$$\star\star$
Gives an explicit description of $\sigma(f_i,i\in I)$ via finite intersections of preimages of sets in ${A'}$.
Proposition 4.6: Komposition messbarer Abbildungen $\star\star$
The composition of two measurable maps is measurable.
Lemma 4.7: Stetigkeit und Urbilder offener Mengen $\star\star$
For metric spaces, a map is continuous iff preimages of open sets are open.
Proposition 4.8: Stetige Funktionen $\mathbb{R}\to\mathbb{R}$ sind Borel-messbar $\star\star$
Every continuous function $f:\mathbb{R}\to\mathbb{R}$ is ${B}(\mathbb{R})$–${B}(\mathbb{R})$-measurable.
Proposition 4.9: Rechenregeln für messbare Funktionen $\star\star\star$
Sums, scalar multiples, products, and suitable quotients of real-valued measurable functions are again measurable.
Lemma 4.12: ${B}(\overline{\mathbb{R}})$ einschränkt sich auf ${B}(\mathbb{R})$$\star\star$
The Borel σ-algebra on $\overline{\mathbb{R}}$ restricts to the usual Borel σ-algebra on $\mathbb{R}$, and $\mathbb{R}$ itself is Borel in $\overline{\mathbb{R}}$.
Korollar 4.13: Messbarkeit nach $\mathbb{R}$ ⇒ Messbarkeit nach $\overline{\mathbb{R}}$$\star\star$
If $f$ is measurable as a map into $\mathbb{R}$, it is also measurable as a map into $\overline{\mathbb{R}}$.
Korollar 4.14: Struktur von ${B}(\overline{\mathbb{R}})$-Mengen $\star$
Any Borel set in $\overline{\mathbb{R}}$ can be written as a Borel subset of $\mathbb{R}$ plus possibly $\pm\infty$.
Korollar 4.15: Charakterisierung der Messbarkeit nach ${B}(\mathbb{R})$$\star\star$
A map $f:\Omega\to\overline{\mathbb{R}}$ is ${A}$–${B}(\overline{\mathbb{R}})$-measurable iff $f^{-1}(B)$, $f^{-1}({-\infty})$, and $f^{-1}({\infty})$ are in ${A}$.
Lemma 4.17: Messbarkeit von Indikatorfunktionen $\star\star$
The indicator $1_A$ is measurable iff $A$ is measurable.
Proposition 4.18: Supremum, Infimum, limsup, liminf sind messbar $\star\star$
Pointwise $\sup$, $\inf$, $\limsup$ and $\liminf$ of sequences of measurable functions are measurable, and the convergence set ${\lim f_n\text{ exists in }\mathbb{R}}$ is measurable.
Korollar 4.20: Einfache Funktionen sind messbar $\star\star$
Every simple function is measurable both as a map into $\mathbb{R}$ and into $\overline{\mathbb{R}}$.
Proposition 4.21: Charakterisierung einfacher Funktionen $\star\star$
A function is simple iff it is measurable and takes only finitely many values, and it admits a representation with disjoint level sets.
Satz 4.22: Approximation nichtnegativer messbarer Funktionen $\star\star\star$
Every nonnegative measurable function can be approximated from below by an increasing sequence of simple functions (uniformly if the function is bounded).
[Script 2024] 5. Lebesgue-Integral
Definitionen
Definition 5.0: Informelle Definition des Integrals $\star\star$
Introduces the integral for simple functions, extends it to nonnegative measurable functions via increasing simple approximations, and then to general measurable functions via positive and negative parts.
Definition 5.2: Lebesgue-Integral für einfache Funktionen $\star\star\star$
For a nonnegative simple function $f=\sum_{i=1}^n\alpha_i 1_{A_i}$, the Lebesgue integral is $\displaystyle \int f,d\mu := \sum_{i=1}^n \alpha_i \mu(A_i)$.
Definition 5.6: Lebesgue-Integral für nichtnegative messbare Funktionen $\star\star\star$
For $f\ge0$ measurable and simple $f_n\uparrow f$, define $\displaystyle \int f,d\mu := \lim_{n\to\infty}\int f_n,d\mu$.
Definition 5.7: Positiv- und Negativteil $\star\star\star$
For measurable $f$, define $f^+=\max(f,0)$ and $f^-=-\min(f,0)$ so that $f=f^+-f^-$.
Definition 5.8: Integrierbar, quasi-integrierbar $\star\star\star$ $f$ is integrable if both $\int f^+,d\mu$ and $\int f^-,d\mu$ are finite.
It is quasi-integrable if at least one is finite; for quasi-integrable $f$ the integral is $\displaystyle \int f,d\mu := \int f^+,d\mu - \int f^-,d\mu$.
Definition 5.9: Räume $L^1$ und $L$$\star\star\star$
Defines $L^1(\Omega,{A},\mu)$ as the set of integrable functions and $L(\Omega,{A},\mu)$ as the set of quasi-integrable functions, with the usual $L^1$-“norm” $|f|_1=\int|f|,d\mu$.
Definition 5.10: Integral über eine Menge $A$$\star\star$
For $f\in L(\mu)$ and $A\in{A}$, the integral over $A$ is $\displaystyle \int_A f,d\mu := \int f\cdot 1_A,d\mu$.
Definition 5.11: Erwartungswert und Varianz $\star\star\star$
On a probability space, the expectation is $EX=\int X,dP$ for $X\in L(P)$ and the variance is $\mathrm{Var}(X)=\int (X-EX)^2,dP$ for $X\in L^1(P)$.
Definition 5.12: Nullmengen und „fast überall“ / „fast sicher“ $\star\star\star$
Defines $\mu$-null sets, “$f\in B$ $\mu$-fast überall” (a.e.), and “$X\in B$ fast sicher” (a.s.) when the complement has measure zero.
Definition 5.20: Bildmaß (Pushforward) $f_H \mu$$\star\star$
For measurable $f:(\Omega,{A})\to(\Omega',{A'})$, the pushforward measure is $(f_H \mu)(A'):=\mu(f^{-1}(A'))$.
(In the PDF notation, the \# is written as a hash symbol.)
Definition 5.30: Dichte $\star\star\star$
If $\nu(A)=\int_A f,d\mu$ as in Proposition 5.29, then $f$ is called the density of $\nu$ with respect to $\mu$ (denoted $d\nu/d\mu=f$).
Theoreme
Lemma 5.1: Darstellung unabhängig für einfache Funktionen $\star\star$
Shows that the integral of a simple function does not depend on which particular simple representation one uses.
Lemma 5.3: Monotonie des Integrals für einfache Funktionen $\star\star$
If $0\le f\le g$ for nonnegative simple $f,g$, then $\int f,d\mu\le\int g,d\mu$.
Lemma 5.4: Monotone Folgen einfacher Funktionen $\star\star$
For $0\le f_1\le f_2\le\cdots\uparrow f$ with a simple $g\le f$, the integrals $\int f_n,d\mu$ form an increasing real sequence and dominate $\int g,d\mu$.
Korollar 5.5: Unabhängigkeit der Approximation in der Definition $\star\star$
For two sequences of simple functions increasing to the same $f\ge0$, the limits of their integrals coincide; thus the integral is well defined.
Lemma 5.13: $\int f,d\mu = 0 \iff f = 0$ a.e. für $f\ge0$$\star\star$
A nonnegative measurable function has integral zero iff it vanishes almost everywhere.
Proposition 5.14: Gleiche Funktionen a.e. ⇒ gleiches Integral $\star\star\star$
If $f\in L(\mu)$ and $g$ is measurable with $f=g$ almost everywhere, then $g\in L(\mu)$ and $\int g,d\mu=\int f,d\mu$.
Proposition 5.15: $L^1$-Funktionen sind fast überall endlich $\star\star$
Any $f\in L^1(\mu)$ is finite almost everywhere and has a real-valued measurable representative with the same integral.
Satz 5.16: Linearität und Monotonie des Integrals $\star\star\star$
The integral is linear on $L(\mu)$ (for well-defined sums and scalar multiples) and monotone in the sense that $f\le g$ a.e. implies $\int f,d\mu\le\int g,d\mu$.
Proposition 5.17: Äquivalente Charakterisierungen von $L^1$$\star\star$
Lists equivalent conditions for $f$ to lie in $L^1(\mu)$, including integrability of $|f|$ and domination by an integrable function.
Korollar 5.18: $\max(f,g)$ und $\min(f,g)$ sind in $L^1$$\star$
If $f,g\in L^1(\mu)$, then $\max(f,g)$ and $\min(f,g)$ are also in $L^1(\mu)$.
Korollar 5.19: Dreiecksungleichung für Integrale $\star\star$
The integral satisfies $|\int f,d\mu|\le\int |f|,d\mu$ for $f\in L(\mu)$.
Proposition 5.21: Bildmaß ist ein Maß $\star\star$
The pushforward $f_H \mu$ is a measure on $(\Omega',{A'})$ with total mass $\mu(\Omega)$, preserving finiteness and probability.
(In the PDF notation, the \# is written as a hash symbol.)
Transformationssatz 5.22: $\int (g\circ f),d\mu = \int g,d(f_H \mu)$$\star\star\star$
For measurable $f$ and $g$, one has $\displaystyle \int_\Omega (g\circ f),d\mu = \int_{\Omega'} g,d(f_H \mu)$,
providing a general change-of-variables formula.
(Again, the \# is a hash symbol in the PDF.)
Lemma 5.23: Vergleich mit einfacher Funktion unter Monotonie $\star\star$
If $0\le g\le\lim f_n$ with $g$ simple and $0\le f_1\le f_2\le\cdots$, then $\int g,d\mu\le\lim_{n\to\infty}\int f_n,d\mu$.
Satz 5.24: Monotone Konvergenz (Beppo-Levi) $\star\star\star$
For $0\le f_1\le f_2\le\cdots$ with $f_n\uparrow f$, one has $\displaystyle \lim_{n\to\infty}\int f_n,d\mu = \int f,d\mu$.
Satz 5.25: Monotone Konvergenz mit Unterabschätzung $\star\star$
If $g\le f_1\le f_2\le\cdots\uparrow f$ and $\int g^-,d\mu<\infty$, then $\lim_n\int f_n,d\mu = \int f,d\mu$
(a monotone convergence theorem for not-necessarily-nonnegative sequences).
Proposition 5.26: Riemann- und Lebesgue-Integral stimmen überein $\star\star$
If $f:[a,b]\to\mathbb{R}$ is Riemann integrable, then it is Lebesgue integrable w.r.t. Lebesgue measure and the two integrals coincide.
Korollar 5.27: Uneigentliche Riemann-Integrale und Lebesgue-Integral auf $[0,\infty)$$\star\star$
For $f\ge0$ Riemann-integrable on $[0,\infty)$ in the improper sense, the improper Riemann integral equals the Lebesgue integral over $[0,\infty)$.
Proposition 5.29: Konstruktion eines Maßes aus einer Dichte $\star\star\star$
Given $f\ge0$ measurable, the map $\nu(A)=\int_A f,d\mu$ defines a measure, and for $g\in L^1(\nu)$ one has $\displaystyle \int g,d\nu=\int fg,d\mu$.
Probability Theory 1 – Exercises collection skeleton
Section 0: Einführung — Aufgaben
Exercise 0.5.1: Basic set identities $\star\star$ done
(a) Distributive law for unions over indexed intersections
(b) De Morgan law for complements of unions.
Exercise 0.5.2: Infinite unions/intersections of open/closed sets $\star\star$ done
(a) Arbitrary unions of open sets are open
(b) Arbitrary intersections of closed sets are closed (via De Morgan).
Exercise 0.5.3: Finite unions/intersections of open/closed sets $\star\star$ done
(a) Finite intersections of open sets are open
(b) Finite unions of closed sets are closed.
Exercise 0.5.4: Validity of algebraic set identities $\star\star$ done
Check which of (a)–(d) hold in general; for those that fail, find sufficient conditions (e.g. inclusions, distributivity, difference rules).
Exercise 0.5.5: Outcome space of an elimination tournament $\star$
Describe the set of all possible results of an $n$-round knockout tournament with $2^n$ players and fixed bracket.
Section 1: Wahrscheinlichkeitsmaß und Wahrscheinlichkeitsraum
Exercise 1.1: Banach–Tarski and non-measurable sets in $\mathbb{R}^3$$\star$
Use the Banach–Tarski paradox to argue that certain subsets of $\mathbb{R}^3$ cannot have a (countably additive) volume.
Exercise 1.2: Properties of σ-algebras $\star\star\star$
(a) A lemma regarding the trivial σ-algebra ${\varnothing,\Omega}$
(b) A lemma regarding the power set ${P}(\Omega)$
(c) A counterexample: an algebra that is not a σ-algebra
(d) A lemma about countable/finite sets and σ-algebras.
Exercise 1.3: Equivalent definitions of algebras and σ-algebras $\star\star$
(a) Replace closure under finite unions by closure under finite intersections in the definition of an algebra
(b) Analogous equivalence for σ-algebras (countable unions vs. countable intersections).
Exercise 1.4: Closure properties of algebras and σ-algebras $\star\star$
(a) An algebra is closed under finite unions, intersections, set difference, and symmetric difference
(b) Every σ-algebra is an algebra.
Show that an algebra on $\Omega$ with only finitely many sets is automatically a σ-algebra.
Exercise 1.6: Countable unions in a σ-algebra $\star\star$
Verify that if $A$ is a σ-algebra and $A_i \in A$, then $\bigcup_{i=1}^\infty A_i \in A$.
Exercise 1.7: Union of σ-algebras need not be a σ-algebra $\star\star$ done
Give an example of σ-algebras $A,B$ on $\Omega$ with $A\cup B$ not a σ-algebra; identify which axiom fails.
Exercise 1.8: Discrete probability measures $\star\star$
(a) Poisson distribution on $\Omega = {0,1,2,\dots}$ defines a probability measure
(b) General discrete probability on a (at most) countable set via weights $p(\omega_i)$.
Exercise 1.9: Algebras on $(0,1]$ via half-open intervals $\star\star\star$ done
Show: $J$ (single intervals $(a,b] \subset (0,1]$) is not an algebra; $J^*$ (finite disjoint unions) is an algebra but not a σ-algebra.
Exercise 1.10: Analogous algebra on $\mathbb{R}$$\star\star$
Define $J_{\mathbb{R}}$ with $(a,b⟩$ and $J_{\mathbb{R}}^*$ as finite disjoint unions; show it is an algebra on $\mathbb{R}$, but not a σ-algebra.
Exercise 1.11: Limits of unions/intersections of intervals $\star\star$ done
Analyze $\bigcup_n(-\infty,t_n]$, $\bigcap_n(-\infty,t_n]$ for monotone sequences $t_n$ with finite or infinite limits.
Exercise 1.12: Independence of interval decomposition $\star\star\star$
Show that for $A\subset(0,1]$ represented as two finite disjoint unions of intervals, the length-based probability $P(A)$ is well defined (independent of representation).
Exercise 1.13: Intersections and unions of σ-algebras $\star\star$ done
(a) Arbitrary intersection of σ-algebras is a σ-algebra
(b) Union of two σ-algebras is usually not a σ-algebra; identify possible axiom violations.
(a) If $A$ is a σ-algebra, then $\sigma(A)=A$
(b) Monotonicity: $M\subset M'\Rightarrow \sigma(M)\subset\sigma(M')$
(c) Stability: $M\subset M'\subset\sigma(M)\Rightarrow\sigma(M')=\sigma(M)$.
Exercise 1.15: Open subsets of $(0,1]$ are Borel $\star\star$
Show that every open set in the subspace $(0,1]$ lies in ${B}((0,1])$.
Exercise 1.16: Monotone class lemma $\star\star\star$ done
(a) Intersection of monotone classes is a monotone class
(b) For an algebra $A$, the smallest monotone class containing $A$ coincides with $\sigma(A)$.
Exercise 1.17: Basic properties of a probability measure $\star\star$
Prove:
(a) $P(\varnothing)=0$
(b) Monotonicity $A\subset B\Rightarrow P(A)\le P(B)$
(c) $P(A)\le1$
(d) $P(\Omega\setminus A)=1-P(A)$
(e) Finite additivity for disjoint sets
(f) Subadditivity for countable unions
(g) Inclusion–exclusion for two sets.
Exercise 1.18: General inclusion–exclusion theorem $\star\star$
Present and prove the general finite inclusion–exclusion formula for probabilities $P(\bigcup_{i=1}^n A_i)$ by induction.
Exercise 1.19: Countable unions of null/full sets $\star\star$ done
(a) Union of countably many null sets is null
(b) Intersection of countably many full-measure sets has probability 1.
Given disjoint intervals $(a_i,b_i⟩$ and initial “thickenings” with parameters $\delta_i$, adjust the $\delta_i$ to keep disjointness while still covering a target interval $(u,v⟩$.
Show identities like $(-\infty,t) = \bigcup_{n}(-\infty,t-1/n)$ and $(-\infty,t] = \bigcap_{n}(-\infty,t+1/n)$.
Exercise 1.22: Continuity from above $\star\star$
If $A_n\downarrow A$, prove $P(A_n)\downarrow P(A)$.
Exercise 1.23: No atoms for the uniform on $[0,1]$$\star\star$ done
For $P$ with cdf $F(x)=0$ for $x<0$, $F(x)=x$ on $[0,1]$, $F(x)=1$ for $x>1$, show $P({x})=0$ for all $x$.
Exercise 1.24: Generators of ${B}(\mathbb{R})$$\star\star\star$ done
Show that each of these families generates the Borel σ-algebra:
(a) $(-\infty,b]$
(b) $(-\infty,b)$
(c) $(a,b)$
(d) $[a,b]$.
Exercise 1.25: Vitali construction of a non-measurable set $\star\star\star$
Construct a Vitali set $M\subset(0,1]$ via the equivalence relation $x\sim y \iff x-y\in\mathbb{Q}$ and show it cannot have a Lebesgue measure.
Exercise 1.26: Trace σ-algebra vs. generated σ-algebra $\star\star$
Show $\sigma(K|M) = \sigma(K)|M$ for $K\subset{P}(\Omega)$, $M\subset\Omega$.
Exercise 1.27: Two descriptions of ${B}((0,1])$$\star\star$
Show that $\sigma(J)$ (from 1.9) coincides with ${(0,1]\cap A : A\in{B}(\mathbb{R})}$.
Exercise 1.28: Trace on $\mathbb{N}$$\star\star$ done
Show $D := {A\cap\mathbb{N} : A\in{B}(\mathbb{R})} = {P}(\mathbb{N})$.
Exercise 1.29: Borel σ-algebra on $[c,d]$$\star\star$
For $0<c<d<1$, show that the two σ-algebras $B_1 := {A\cap[c,d] : A\in{B}((0,1])}$ and $B_2 := {B}((c,d]) \cup {A\cup{c}: A\in{B}((c,d])}$
coincide.
Exercise 1.30: Restriction of Lebesgue measure and cdf $\star\star$
(a) Re-derive ${A\cap(0,1] : A\in{B}(\mathbb{R})} = {B}((0,1])$
(b) Define $\mu(B) := \lambda(B\cap(0,1])$ and show it is a probability on $(\mathbb{R},{B}(\mathbb{R}))$ with the given cdf and that $\mu(C)=\lambda(C)$ for $C\subset(0,1]$
(c) Given a probability $\mu$ with that cdf, show its restriction to $(0,1]$ equals $\lambda$.
Exercise 1.31: Probability measures on bounded intervals $\star\star$
(Conceptual) Construction of uniform probability measures on $(a,b]$, on $[0,1]$, and analogs on other bounded intervals via restriction and renormalization.
Exercise 1.32: Decomposition of a cdf $\star\star$
Given a cdf $F$ with standard properties, decompose $F = c_1F_1 + c_2F_2$ into a continuous part and a purely atomic part $F_2(x) = \sum_i \alpha_i 1_{[z_i,\infty)}(x)$.
Exercise 1.33: Lebesgue measure on $\mathbb{R}$$\star\star\star$
Construct the global Lebesgue measure $\lambda$ on $(\mathbb{R},{B}(\mathbb{R}))$ from restrictions to intervals $\Omega_n=(n-1,n]$:
(a) Show $\lambda$ is σ-finite and satisfies the measure axioms
(b) Show $\lambda((a,b])$ equals the length $b-a$.
Section 2: Messbare Abbildungen und Zufallsvariable
Exercise 2.1: Basic properties of preimages $\star\star\star$ done
Show that $T^{-1}$ preserves unions, intersections, and set differences.
(a) Any constant map is measurable
(b) A finitely-valued $T:\Omega\to\mathbb{R}$ is measurable iff all singletons’ preimages are measurable.
Exercise 2.4: Joint σ-algebra generated by a family of maps $\star\star$ done
Given a family ${f_i}$, show “all $f_i$ are measurable” is equivalent to $M\subset A$, where $M$ is the preimage system and $\sigma(M)$ is the σ-algebra generated by the family.
Exercise 2.5: Monotone functions are Borel-measurable $\star\star$
Show that every non-decreasing $f:\mathbb{R}\to\mathbb{R}$ is ${B}(\mathbb{R})$-measurable, via the structure of $f^{-1}((-\infty,b])$.
Use that preimages of open sets are open and that open sets are Borel to prove every continuous $f:\mathbb{R}\to\mathbb{R}$ is Borel-measurable.
Exercise 2.7: Images of measurable sets $\star\star$ done
(a) Construct a measurable map whose image of a measurable set is not measurable in the target σ-algebra
(b) Show that a continuous map may send open sets to non-open sets.
Exercise 2.8: Algebraic operations on measurable functions $\star\star\star$
Given $T_1,T_2$ measurable:
(a) Show measurability of the product $T_1T_2$
(b) If $T_2\neq0$, show measurability of $T_1/T_2$ via composition with $x\mapsto1/x$.
Exercise 2.9: Intersection σ-algebra on a subset $\star\star$
Show ${B\cap A : A\in{A}} = {A\in{A} : A\subset B}$.
Exercise 2.10: Measurability via restriction to a partition $\star\star$
Given a countable cover $\Omega=\bigcup A_n$, show $T$ is measurable iff each restriction $T_n$ to $A_n$ is measurable.
Recall and prove:
(a) Proposition 2.1 (approximation of sup/inf by elements of the set)
(b) Lemma 2.2 (limit of a monotone sequence = sup/inf)
(c) Proposition 2.3 (definitions and formulas for lim inf and lim sup).
Exercise 2.12: Measurable sets of pointwise convergence $\star\star$
For measurable $T_n,T$, show the set ${\omega : T_n(\omega)\to T(\omega)}$ lies in ${A}$.
Exercise 2.13: Borel σ-algebra on extended reals $\star\star$
Show ${B}(\overline{\mathbb{R}})$ is generated by $(-\infty,b]$ plus the singletons ${-\infty}$ and ${\infty}$.
Exercise 2.14: Trace relation of Borel σ-algebras $\star\star$
Show ${B}(\mathbb{R})$ is the trace σ-algebra of ${B}(\overline{\mathbb{R}})$ on $\mathbb{R}$.
Exercise 2.15: Measurability via $\mathbb{R}$ vs. $\overline{\mathbb{R}}$$\star\star$
Show $T:\Omega\to\mathbb{R}$ is ${A}$–${B}(\mathbb{R})$ measurable iff it is ${A}$–${B}(\overline{\mathbb{R}})$ measurable.
Exercise 2.16: Measurable max and min $\star\star$ done
Show that if $S,T$ are extended-real-valued measurable maps, then $\max(S,T)$ and $\min(S,T)$ are measurable.
Exercise 2.17: Reciprocal and log of a measurable function $\star\star$ done
(a) Show the extended-valued map $x\mapsto1/f(x)$ (with $\infty$ at zeros) is measurable
(b) For $f\ge0$, show the extended-valued $\log f$ (with $-\infty$ at zeros) is measurable.
Exercise 2.18: Arithmetic with extended-real measurable functions $\star\star\star$ done
Using truncations $S_n,T_n$:
(a) Show measurable $S,T$ ⇒ measurable $S+T$ (when well-defined)
(b) Show measurability of $ST$
(c) Show measurability of $S/T$ where defined.
Exercise 2.19: Pathology with non-measurable sets $\star\star$
Take a non-measurable $A$, define $f=1_A, g\equiv1$, and verify that although $f,g$ are measurable into the trivial σ-algebra ${\varnothing,\mathbb{R}}$, the set ${f=g}=A$ is not Borel-measurable.
Exercise 2.20: Egorov’s theorem $\star\star\star$
On a probability space, show that pointwise a.s. convergence of $T_n\to T$ implies: for every $\eta>0$ there exists $A$ with $P(A)<\eta$ such that $T_n\to T$ uniformly on $A^c$.
Exercise 2.21: σ(T) and measurability $\star\star$
(a) Show $\sigma(T^{-1}(A'))$ is a σ-algebra on $\Omega$ (the σ-algebra generated by $T$)
(b) Show $T$ is measurable w.r.t. ${A},A'$ iff $\sigma(T)\subset{A}$.
Exercise 2.22: W-transformation of a cdf $\star\star$
For a strictly increasing, continuous cdf $F$ of $X$, show $F(X)$ has the standard uniform-type distribution $G(x)=0$ for $x\le0$, $G(x)=x$ for $0<x<1$, $G(x)=1$ for $x\ge1$ (the W-transform).
Exercise 2.23: Restriction of a measurable map $\star\star$
Show that if $T$ is measurable on $(\Omega,{A})$, then its restriction $T|_{\Omega_0}$ is measurable on $(\Omega_0,{A}\cap\Omega_0)$.
Exercise 2.24: Approximation by simple functions $\star\star\star$
Show any extended-real measurable $T$ can be approximated pointwise by simple measurable $T_n$, with monotone approximation for $T\ge0$ or $T\le0$, and uniform convergence if $T$ is bounded.
Section 3: Das Lebesgue-Integral
Exercise 3.1: Independence of simple-function representation $\star\star\star$
Show the Lebesgue integral of a simple function is independent of the particular finite sum representation.
Exercise 3.2: Structure of simple functions and their integrals $\star\star$
On a probability space:
(a) Show the simple functions form a vector space
(b) Show the map $T\mapsto\int T,dP$ is linear
(c) Show any simple $T$ can be written using a partition of $\Omega$
(d) On $\Omega=\mathbb{N}$ with counting σ-algebra, show the indicator family ${1_{{n}}}$ is linearly independent.
Exercise 3.3: Integrability of $\exp$ on $(0,1]$$\star\star$
On $((0,1],{B},\lambda)$: show $\exp$ is measurable, the Lebesgue integral $\int_0^1 e^x,d\lambda$ exists, and compute it via simple-function approximation.
Exercise 3.4: Integrals of indicators of rationals/irrationals $\star\star\star$
On $(0,1]$ with Lebesgue measure, show $\int 1_{\mathbb{Q}\cap(0,1]}d\lambda=0$ and $\int 1_{(\mathbb{R}\setminus\mathbb{Q})\cap(0,1]}d\lambda=1$.
Exercise 3.5: Integral as supremum over simple minorants $\star\star\star$
For $S\ge0$ measurable, show $\int S,dP = \sup{\int T,dP : 0\le T\le S,\ T\ \text{simple}}$.
Exercise 3.6: Integrals on a countable probability space $\star\star$
On $\Omega=\mathbb{N}$ with $P({n})=p_n$, compute $\int T,dP$ for $T\ge0$ in terms of $\sum T(n)p_n$.
Exercise 3.7: Failures of monotone convergence when assumptions are dropped $\star\star$
Provide counterexamples showing:
(a) Without $T_n\ge0$, monotone convergence can fail
(b) Without monotonicity, it can fail
(c) For decreasing $T_n$, even with $T_n\ge0$, the naive “downward” version can fail.
Exercise 3.8: Relation Riemann vs. Lebesgue integral on $[a,b]$$\star\star\star$
(a) If $f:[a,b]\to[0,\infty)$ is Riemann-integrable, then it is Lebesgue-integrable with the same value
(b) Give a function Lebesgue-integrable but not Riemann-integrable
(c) Give $f:\mathbb{R}\to\mathbb{R}$ that is improperly Riemann-integrable but not Lebesgue-integrable.
Exercise 3.9: Measurability and decomposition into positive/negative parts $\star\star$
(a) For extended-real measurable $S,T$, prove measurability of $S+T$ when well-defined
(b) Show $T$ is measurable iff its positive and negative parts $T^+,T^-$ are measurable.
Exercise 3.10: $M$ and $L^1(P)$ as vector spaces $\star\star$
(a) $M$: the space of real-valued measurable functions, is a vector space
(b) $L^1(P)$ is a vector space, and $\int T,dP$ is linear
(c) Show $L^1(P)$ is infinite-dimensional
(d) Show $|T|_1 := \int|T|,dP$ is a seminorm on $L^1(P)$.
Exercise 3.11: $L^p$ spaces and inner product on $L^2$$\star\star$
(a) For $1\le p<\infty$, show $L^p(P)$ is a vector space and $|\cdot|_p$ a seminorm
(b) On $L^2(P)$, show $\langle S,T\rangle:=\int ST,dP$ is a semi-inner product and $|T|_2^2 = \langle T,T\rangle$.
Prove Cauchy–Schwarz in an abstract semi-inner-product space using the quadratic polynomial argument.
Exercise 3.13: Finite probability spaces and $\ell^1$$\star\star$
On a finite probability space, identify $L^1(P)$ with $\mathbb{R}^{|\Omega|}$ via a norm-preserving linear isomorphism.
Exercise 3.14: Densities and change of measure $\star\star\star$
(a) If $T_0\ge0$ with $\int T_0,dP=1$, define $Q(A)=\int T_0 1_A,dP$ and show $Q$ is a probability measure absolutely continuous w.r.t. $P$
(b) Show for $T\ge0$: $\int T,dQ = \int TT_0,dP$.
Exercise 3.15: Product of integrable/quasi-integrable functions $\star\star$
(a) Show measurability of $T_1T_2$ for extended-real measurable $T_1,T_2$
(b) Example: integrable $T_1,T_2$ with non-integrable product
(c) Example: quasi-integrable $T_1,T_2$ with product not quasi-integrable.
For measurable $T$ and scalar $\alpha$:
(a) Show measurability of $\alpha T$
(b) If $T$ is integrable, so is $\alpha T$, and $\int\alpha T,dP = \alpha\int T,dP$
(c) Analogous statement for quasi-integrability.
Exercise 3.17: Linearity of the integral (extended-real case) $\star\star$
For measurable $T_1,T_2$ with $T_1+T_2$ well-defined:
(a) Show measurability of $T_1+T_2$
(b) If both integrable, then $\int(T_1+T_2),dP = \int T_1,dP + \int T_2,dP$
(c) Extend to the case “one integrable, one quasi-integrable”.
If $T_n\uparrow T$ with $T_n\ge S$ and $\int S^-,dP<\infty$, prove $\int T_n,dP\to\int T,dP$; similarly for $T_n\downarrow T$ dominated by $S^+$.
Exercise 3.19: A mixed measure on $(0,1]$$\star\star$
On $(0,1]$ with $\lambda$:
(a) Show $P := \frac12\lambda + \frac14\delta_a + \frac14\delta_b$ is a probability measure
(b) For $T\equiv1$, compute $\int_{[a,b]} T,dP$ and $\int_{(a,b)} T,dP$, illustrating the subtlety of $\int_a^b$ notation.
