Measure

Measure is a mathematical concept that generalizes the everyday notion of size. Intuitively one can think of measure as length, weight, volume, distance from a point, probability, etc.

Consider a non-empty set $\fs4 \Omega$ . Also consider a $\fs4 \sigma$ -algebra on $\fs4 \Omega$ , denoted with $\fs4 \mathscr{F}$ . The pair $\fs4 (\Omega, \mathscr{F})$ is called a measure space, since we can apply the measure function on the members of $\fs4 \mathscr{F}$ . The members of $\fs4 \mathscr{F}$ are called just measurable.

Contrary to intuition, not all subsets of $\fs4 \Omega$ are (necessarily) measurable, and therefore there are subsets that don't belong in any $\fs4 \sigma$ -algebra $\fs4 \mathscr{F}$ . An important example of such subsets is the Vitali set.

A measure $\fs4 \mu$ will be a function

$\fs4 \mu : \mathscr{F} \rightarrow [0, +\infty]$

which satisfies the following properties:

The empty set has measure zero

$\fs4 \mu(\emptyset) = 0$

The measure of a countable union of disjoint sets equals the sum of their measures. If $\fs4 F_1, F_2, \ldots \in \mathscr{F}$ with $\fs4 F_i \cap F_j = \emptyset$ for $\fs4 i \neq j$ , then

$\fs4 \mu(\cup_{i=1}^\infty F_i) = \sum_{i=1}^\infty \mu(F_i)$

We are particularly interested in measures that have the extra property that the total measure of $\fs4 \Omega$ is one

$\fs4 \mu(\Omega) = 1$

If we look at the members of $\fs4 \mathscr{F}$ as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

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Measure