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;

$\mu \left( \cup_i^\infty A_i \right) = \sum_i^\infty \mu (A_i)$

where $A_i$ are disjoint measurable sets. Additivity makes sense because we want $\mu(A) + \mu(B) = \mu(A \cup B)$ 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, $\mu(E) = 0, \forall x \in E^{c}, P(x)$.

A function f from a measurable space (X, F) to a measurable space (Y, G) is measurable if $\forall E \in \mathcal{G}, f^{-1}(E) \in \mathcal{F}$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): $\int \mathbb{I}_{A} \mathrm{d}\mu = \mu(A)$.

References:

1. Halmos, P. R. Measure theory. Springer-Verlag, 1974
2. 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.

One Comment leave one →