Lebesgue Measurability

Definition §

A function $f:[a,b]\to \mathbb{R}$ is measurable if for some sequence of continuous functions $\{f_n{\}}_{n=1}^{\infty }$, $f_n(x)\underset{n\to \infty }{\to }f(x)$ a.e.

A function over an infinite interval, $f:\mathbb{R}\to \mathbb{R}$, is measurable if it is measurable over all possible finite intervals.

A subset $A$ of $\mathbb{R}$ is measurable if and only if its indicator function, $1_A$, is measurable.

Properties §

1. Measurability of functions is closed under linear combination ($f,g$ measurable $⟹$ $\alpha f+\beta g$ measurable)
2. If a sequence of measurable functions $\{f_n{\}}_{n=1}^{\infty }\underset{n\to \infty }{\to }f$, then $f$ is also measurable.
3. For a sequence of measurable functions $\{f_n{\}}_{n=1}^{\infty }$, $\underset{n}{\sup }{f}_n$ and $\underset{n}{\inf }{f}_n$ are also measurable.
4. Measurability of sets is closed under reasonable set operations (union, intersection, difference, complement, etc.)

Remarks §

1. Since convergence is only necessary a.e., $f$ itself need not be continuous.

Examples §

1. On the interval $[0,1]$, consider the sequence $f_n(x)=x^n\underset{n\to \infty }{\to }f(x)=\{\begin{array}{l}0,\text{ if }0\le x<1\\ 1,\text{ if }x=1\end{array}$. Here, $f(x)$ is measurable.