A Measure Theory Tutorial (Measure Theory for Dummies)

This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding. A Something to measure First, we need something to measure. So we define a “measurable space.” A measurable space is a collection of events B, and the set of all outcomes Ω, which is sometimes called the sample space. Given a collection of possible events B, why do you need to state Ω? For one, having a sample space makes it possible to define complements of sets; if the event F ∈ B, then the event F is the set of outcomes in Ω that are disjoint from F . A measurable space is written (Ω,B). A.1 Algebras and Fields Often, you will see that the collection of events B in a measurable space is a σ-algebra. A σ-algebra is a special kind of collection of subsets of the sample space Ω: a σ-algebra is complete in that if some set A is in your σ-algebra, then you have to have A (the complement of A) in your set too. Also, it must be that if you have two sets A and B in your collection of sets, then the union A ∪ B must also be in your collection of sets (in fact, σ-algebras are closed under countable unions, not just finite unions). Another term sometimes used to mean the same thing as σ-algebra is σ-field. The smallest possible σ-field is a collection of just two sets, {Ω, ∅}. The largest possible σ-field is the collection of all the possible subsets of Ω, this is called the powerset. B Measure Ameasure μ takes a setA (from a measurable collection of sets B), and returns “the measure ofA,” which is some positive real number. So ones writes μ : B → [0,∞). An example measure is volume, which goes by the name Lebesgue measure. In general, measures are generalized notions of volume. The triple (Ω,B, μ) combines a measurable space and a measure, and thus the triple is called a measure space. A measure is defined by two properties: