Measures
Why
We extend our notion of length, area, and volume beyond the Lebesgue measure on the product spaces of real numbers.
Definition
Suppose $\mathcal{A} $ is an algebra of sets.
A function $f: \mathcal{A} \to \Rbar_+$ is
finitely additive if
\[
f(\cup_{i = 1}^{n} A_i) = \sum_{i = 1}^{n} f(A_i) \quad
\text{for all disjoint } A_1, \dots , A_n \in \mathcal{A}
\]
Similarly, suppose $\mathcal{F} $ is a
$\sigma $-algebra.
Then $f$ is countably
additive if
\[
f(\cup_{i = 1}^{\infty} F_i) = \sum_{i =1 }^{\infty} f(F_i)
\quad \text{for all disjoint sequences } \set{F_i}_{i \in \N }
\text{ in } \mathcal{F}
\]
If, in addition, $f(\varnothing) = 0$, then $f$ is called a finitely additive measure or countably additive measure respectively. Since a countably additive measure is finitely additive (the converse is false!), when we speak of a measure we mean a countable additive one.
When $(X, \mathcal{F} )$ is a countably unitable subset algebra and $\mu : \mathcal{F} \to \Rbar_+$, then we call $(X, \mathcal{F} )$ a measurable space and call $(X, \mathcal{F} , \mu )$ a measure space. The word “space” is natural, since the notion of a measure generalized the notion of volume in real space (see Real Space and N-Dimensional Space). We often call $\mathcal{F} $ the measurable sets. In other words, a measure space is a triple: a base set, a sigma algebra, and a measure.
Notation
We often use $\mu $ for a measure since it is a mnemonic for “measure”. We often also us $\nu $ and $\lambda $ since these letters are near $\mu $ in the Greek alphabet.
Examples
Let $(A, \mathcal{A} )$ a measurable space. Let $\mu : \mathcal{A} \to [0, +\infty]$ be $0$ if $A = \varnothing$ and $\mu (A)$ is $+\infty$ otherwise. Then $\mu $ is a measure.
\[ 1 = \mu (B \cup C) \neq \mu (B) + \mu (C) = 2 \]
Properties
\[ \mu (B) \leq \mu (C) \quad \text{for all } B \subset C \subset A \]
\[ m(\cup_{n = 1}^{\infty} A_i) = \lim_{n \to \infty} m(A_i) \]
\[ m(\cap _{n = 1}^{\infty} A_i) = \lim_{n \to \infty} m(A_i) \]