Program.cs

namespace MissingProperty
{
    using System;
    using System.Linq;
    using System.Security;
    using System.Threading.Tasks;

    using Microsoft.Azure.Documents;
    using Microsoft.Azure.Documents.Client;

    using Newtonsoft.Json;

    class Program
    {
        const string url = "https://[YourDocDbAccoubtName].documents.azure.com:443/";
        const string key = "[YourKey]";

        const string DbName = "MissingPropertyTest";

        const string CollectionName = "Default";

        static void Main(string[] args)
        {
            // This will cause DocumentDb to not store "null" values in the database as an example
            JsonConvert.DefaultSettings = () => new JsonSerializerSettings
                                                    {
                                                        NullValueHandling = NullValueHandling.Ignore
                                                    };


            var client = GetClient();
            InitializeDb(client);

            var item1 = new MyItem() {Id = "1", SomeValue = null };
            var item2 = new MyItem() { Id = "2", SomeValue = "abc" };
            var item3 = new MyItem() { Id = "3" };
            var collectionUrl = UriFactory.CreateDocumentCollectionUri(DbName, CollectionName);
            client.CreateDocumentAsync(collectionUrl, item1).Wait();
            client.CreateDocumentAsync(collectionUrl, item2).Wait();
            client.CreateDocumentAsync(collectionUrl, item3).Wait();

            Console.WriteLine("Naive search");
            var query1 = client.CreateDocumentQuery<MyItem>(collectionUrl).Where(i => i.SomeValue == null);
            Console.WriteLine(query1.ToString());
            foreach (var myItem in query1)
            {
                Console.WriteLine(myItem);
            }

            Console.WriteLine("Check for undefined property");
            var query2 = client.CreateDocumentQuery<MyItem>(collectionUrl).Where(i => (i.SomeValue ?? null) == null);
            Console.WriteLine(query2.ToString());
            foreach (var myItem in query2)
            {
                Console.WriteLine(myItem);
            }

            Console.WriteLine("Done");
            Console.ReadKey();
        }

        private static DocumentClient GetClient()
        {
            return new DocumentClient(new Uri(url), GetAuthKey(), new ConnectionPolicy() {ConnectionMode = ConnectionMode.Direct});
        }

        private static void InitializeDb(DocumentClient client)
        {
            CreateDatabaseIfNotExists(client).Wait();
            CreateCollectionIfNotExists(client).Wait();
        }

        private static async Task CreateDatabaseIfNotExists(DocumentClient client)
        {
            if (!client.CreateDatabaseQuery()
                .Where(db1 => db1.Id == DbName)
                .ToList().Any())
            {
                await client.CreateDatabaseAsync(new Database { Id = DbName }).ConfigureAwait(false);
            }
        }

        private static async Task CreateCollectionIfNotExists(DocumentClient docClient)
        {
            if (!docClient.CreateDocumentCollectionQuery(UriFactory.CreateDatabaseUri(DbName)).Where(c => c.Id == CollectionName).ToList().Any())
            {
                var docCollection = new DocumentCollection()
                {
                    Id = CollectionName, 
                };

                await docClient.CreateDocumentCollectionAsync(UriFactory.CreateDatabaseUri(DbName), docCollection).ConfigureAwait(false);
            }
        }

        private static SecureString GetAuthKey()
        {
            var results = new SecureString();
            foreach (var ch in key)
            {
                results.AppendChar(ch);
            }

            return results;
        }
    }

    public class MyItem
    {
        [JsonProperty("id")]
        public string Id { get; set; }

        public string SomeValue { get; set; }

        public override string ToString()
        {
            return $"{this.Id} - {this.SomeValue ?? "NULL"}";
        }
    }
}

See http://blog.lytzen.name/2016/09/find-documents-with-missing-properties.html for background