# An engineering introduction to measure theory

A measure is a convenient mathematical object (function) that can represent the positions of strawberries in a field, distribution of water in the ocean, or probabilities of winning over the lottery numbers — the measure counts the number of strawberries in a given area, reports the amount of water in a certain sea, and evaluates the probability of a lottery ticket to win. This abstract unifying framework enables one to rigorously ‘measure’ quantities over a space, and also enable integration. It also allows elegant notation for probability theory. Here we briefly describe key ideas of measure theory without proof. This material is mostly based on references [1] and [2].

To define a measure, we need a **measurable space** (X, F); a non-empty set X and a σ-algebra F on X. Here X is the space where our stuff to be measured lies, and F gives special structure of the space such that things are well defined and pathological sets can be avoided. An **algebra** F of X is a set of subsets of X such that it contains the empty set, and closed under set union and complement. A **σ-algebra** is an algebra that is closed under countable union. Elements in F are said to be **measurable**.

A **measure** μ on (X, F) is a non-negative extended real valued function on F that is countably additive;

where are disjoint measurable sets. Additivity makes sense because we want when A and B are disjoint (number of strawberries should add up for different fields). If μ is always finite valued, then it is called a **finite measure**. For a special case, a probability measure is a finite measure where μ(X) = 1.

Given a set E, we denote the smallest σ-algebra that contains E as **σ(E)** and say σ(E) is the σ-algebra generated by E. A measure μ on F is determined by its values on any algebra that generates F (Carathéodory extension theorem).

A predicate P(x) holds **almost everywhere μ** (or **μ-a.e.** for short) if it is true except for a set of measure zero, that is, .

A function f from a measurable space (X, F) to a measurable space (Y, G) is **measurable** if . For a topological space (X, U), the σ-algebra generated by the open sets is called the **Borel σ-algebra** (or Borel set). Borel set links the measurable functions and continuous functions — every continuous function from X to the real line is measurable with respect to the Borel algebra. When the topological space is induced by a metric, Borel set and Baire set coincide. **Baire set** is the smallest σ-algebra with respect to which the continuous functions are measurable.

Real-valued Borel-measurable functions are closed under algebraic operations and limit. Moreover, they can be approximated by limit of simple functions. A simple function is a finite linear combination of indicator function of measurable sets. By the linearity of integration, integration of a Borel-measurable function with respect to a measure μ can be defined by letting the integration of an indicator function on the measurable set A as μ(A): .

References:

- Halmos, P. R. Measure theory. Springer-Verlag
*,*1974 - Daley, D. J. & Vere-Jones, D. An Introduction to the Theory of Point Processes. Springer, 1988

P.S. Terry Tao has a great summary on measure theory and integration.

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