By definition, a sequence of functions $f_n$ converges to $f$ almost everywhere in a set $D$ if $m(\left\{x\in D \text{ such that } f_n(x) \text{ does not converge to } f(x)\right\}) = 0$ where $m$ is the Lebesgue measure. My question here is how do we relate this kind of definition to uniform convergence?
Can we show by example that almost everywhere convergence doesn't imply uniform convergence? In other words, an example of a sequence of functions that converges almost everywhere but does not converge uniformly, and why it fails to converge uniformly.
There are sequences of functions that converge everywhere (not just almost) but not uniformly. For instance on the interval (0,1) take $f_n = \min(n,1/x)$. $f_n$ converge to $1/x$ everywhere, but not uniformly.