General Measurable Spaces and Measure Spaces

Definition: Let $X$ be a set and let $\mathcal A$ be a $\sigma$-algebra over $X$. Then the pair $(X, \mathcal A)$ is called a Measurable Space. If $E \in \mathcal A$ then $E$ is said to be a Measurable Set

Consider the set $\mathbb{R}$ of real numbers. We have already proven that the set of all Lebesgue measurable sets, $\mathcal M$, is a $\sigma$-algebra on $\mathbb{R}$. Therefore $(\mathbb{R}, \mathcal M)$ is a measurable space.

For another example, let $X$ be any set. Then the power set $\mathcal P(X)$ is $\sigma$-algebra on $X$ and therefore $(X, \mathcal P(X))$ is a measurable space.

Definition: A Measure on a measurable space $(X, \mathcal A)$ is a function $\mu : \mathcal A \to [0, \infty]$ that has the following properties: 1) $\mu (\emptyset) = 0$. 2) For every countable collection of mutually disjoint measurable sets $(E_n)_{n=1}^{\infty}$ we have that $\displaystyle{\mu \left ( \bigcup_{n=1}^{\infty} E_n \right ) = \sum_{n=1}^{\infty} \mu (E_n)}$ (countable additivity).

We've already equipped the measurable space $(\mathbb{R}, \mathcal M)$ with the Lebesgue measure $m : \mathcal M \to [0, \infty]$ defined for all Lebesgue measurable sets $E \in \mathcal M$ by:

(1)

\begin{align} \quad m(E) = \inf \left \{ \sum_{n=1}^{\infty} l(I_n) : E \subseteq \bigcup_{n=1}^{\infty} I_n \: \mathrm{and} \: \{ I_n = (a_n, b_n) \}_{n=1}^{\infty} \right \} \end{align}

We noted that $m(\emptyset) = 0$, and on The Lebesgue Measure of a Countable Union of Mutually Disjoint Lebesgue Measurable Sets page we proved that if $(E_n)_{n=1}^{\infty}$ is a countable collection of mutually disjoint Lebesgue measurable sets then:

(2)

\begin{align} \quad m \left ( \bigcup_{n=1}^{\infty} E_n \right ) = \sum_{n=1}^{\infty} m(E_n) \end{align}

Therefore $m$ is a measure on $\mathcal M$.

Definition: If $X$ is a set, $\mathcal A$ is a $\sigma$-algebra on $X$, and $\mu : \mathcal A \to [0, \infty]$ is a measure on $\mathcal A$, then the triple $(X, \mathcal A, \mu)$ is called a Measure Space.

The measure space that we have been working with so far is $(\mathbb{R}, \mathcal M, m)$ of Lebesgue measurable sets.

Definition: If $(X, \mathcal A, \mu)$ is a measure space then a property is said to hold $\mu$-almost everywhere on a measurable set $E$ if there exists a measurable set $E_0 \subseteq E$ such that $\mu (E_0) = 0$ and the property holds on $E \setminus E_0$.