Pointwise convergence vs. almost sure convergence I do not understand the difference between these two types of convergence for random variables. Actually, I am not seeing a lot of people using the notion of pointwise convergence for random variables. 
So, what is the difference? Does anyone have a memorable example illustrating the difference? 
 A: Let $\{f_n\}$ be a sequence of measurable functions $f_n:X\to\mathbb R$, where $X$ is some measure space with measure $\mu$. Let $f:X\to \mathbb R$ be another measurable function. 
The sequence $\{f_n\}$ converges to $f$ pointwise if for each $x\in X$, the sequence of real numbers $f_n(x)$ converges to $f(x)$. Notice that this doesn't use the measure $\mu$ at all.
The sequence $\{f_n\}$ converges to $f$ almost everywhere (or almost surely, or pointwise almost everywhere, etc.) if there is a subset $B\subset X$ with $\mu(B)=0$ such that $\{f_n\}$ converges to $f$ pointwise on $X\setminus B$. Observe that pointwise convergence implies a.e. convergence by taking $B=\varnothing$.
An example to illustrate the difference. Let $f_n(x)=x^n$ on the unit interval $[0,1]$ with the standard Lebesgue measure. Then $f_n\to 0$ almost everywhere since, if $x\neq 1$, then $x^n\to 0$ as $n\to\infty$, so that we can take $B=\{1\}$ in the definition of a.e. convergence. However, $f_n\not\to 0$ pointwise since $f_n(1)=1\neq 0$ for each $n$.
