1  Measurability

Published

October 26, 2025

1.1 Special kinds of set collections

1.1.1 p-system

A collection of sets that is closed under intersection.

1.1.2 d-system

A collection of sets \mathcal D that:

  1. contains the universal set E.
  2. is closed under subset subtraction i.e. if A, B \in \mathcal D such that B \subset A then B \setminus A \in \mathcal D.
  3. is closed under monotone limits i.e. if (A_n) \subset \mathcal D, A_n \nearrow A then A \in \mathcal D

1.1.3 Sigma algebra

A collection of sets that is:

  1. non empty
  2. closed under complementation
  3. closed under countable union

1.1.3.1 Properties of sigma algebras

  1. A collection of sets is a sigma algebra if and only if it is both a p-system and a d-system.
  2. The intersection of an arbitrary (even uncountable) collection of sigma algebras is a sigma algebra.
  3. A sigma algebra always contains the null set and the universal set.
  4. A sigma algebra is closed under countable union, countable intersection, set difference.
  5. The intersection of all sigma algebras containing \mathcal C is the smallest sigma algebra containing \mathcal C.

1.2 Monotone class theorem

If \mathcal D is a d-system that contains a p-system \mathcal C, then \mathcal D actually contains all of \sigma \mathcal C.

1.2.1 Good sets technique

Let’s say we want to prove some condition for a sigma algebra \mathcal F. But since proving everything for every element of a sigma algebra is impossible, we can reduce the problem to finding p-system \mathcal C that generates that sigma-algebra, proving that property for \mathcal C, and then proving that the collection of good sets.

Steps:

Define the collection of all good sets \mathcal G = \{B \subset E : B \text{ satisfies the condition}\}.

  1. Prove \mathcal G is a d-system.
  2. Find a generator \mathcal C and prove that \sigma \mathcal C = \mathcal F
  3. Prove that \mathcal C is a p-system.
  4. Prove that \mathcal C \subset \mathcal G i.e. all members of \mathcal C satisfy the condition.

By the monotone class theorem, \sigma \mathcal C = \mathcal F \subset \mathcal G and hence the condition holds for all sets in \mathcal F.

1.2.2 Example

Given two set functions, f and g, if they agree on all sets A \in \mathcal C, they agree for all sets in \sigma \mathcal C.

1.2.3 Simpler but less general form of good sets technique

Define the collection \mathcal A = \{ B \in \mathcal F : B \text{ satisfies the condition} \}

  1. Prove \mathcal A is a sigma algebra.
  2. Find a generator \mathcal C and prove that \sigma \mathcal C = \mathcal F
  3. Prove that \mathcal C \subset \mathcal A i.e. all members of \mathcal C satisfy the condition (they are in the sigma algebra by step 2).

Since \mathcal A is a sigma algebra it is also a d-system. Hence by the monotone class theorem \mathcal C \subset \mathcal A implies \sigma C = \mathcal F \subset \mathcal A.

This implies \mathcal A = \mathcal F and thus the condition holds for all members of \mathcal F.

This can be used for example to prove that a function is measurable if the preimages of all sets in a generator collection belong to the domain sigma algebra.

1.2.4 Borel sigma algebra

1.2.4.1 Examples of Borel sets

  1. Singletons
  2. [a, b], (a,b), (a,b] and [a,b) where a, b \in \bar{\mathbb R}

1.3 Measurable functions

1.3.1 Operations that preserve measurability

  1. Supremum, infimum of countable collections of functions
  2. Limit superior, limit inferior of countable sequences of functions
  3. Composition of measurable functions
  4. Concatenation (“product”) of measurable functions

1.3.2 Useful results on measurability

  1. Continuous functions are measurable: a function f: S \rightarrow T between two topological spaces (S,\tau_S) and (T,\tau_T) is continuous iff for every A \in \tau_T, f^{-1} A \in \tau_S.

  2. A function f: E \rightarrow \bar{\mathbb R} is measurable iff f^+(x) = \max (0, f(x)) and f^-(x) = \max (0, -f(x)) are both measurable.

  3. Simple functions are measurable: a function f: E \rightarrow \bar{\mathbb R} is simple iff it can be expressed in the form \sum_{i=1}^n a_i \mathbb 1 (x \in A_i) where a_i \in \mathbb R, n is finite and (A_i)_{i=1}^n \subset \mathcal E.

  4. Any non-negative function f: E \rightarrow \mathbb R_+ is measurable iff it is the pointwise limit of a monotone increasing sequence of simple functions.

1.3.3 Techniques for proving measurability

Non-obvious ways of proving a function f: E \rightarrow F is \cal E \backslash F measurable.

  1. Show that f is the limit of a sequence of simple functions.
  2. Show that for all sets A \in \mathcal C such that \sigma \mathcal C = \mathcal F, f^{-1} A \in \mathcal E.

1.4 Monotone class theorem for functions