Definition:Almost Uniform Convergence

Given a measure space $$(X, \Sigma, \mu)\ $$ and a sequence of $\Sigma $-measurable functions $$f_n:D\to\R$$ for $$D\in\Sigma$$, $$f_n\ $$ is said to converge uniformly almost everywhere (or converge uniformly a.e.) on $$D\ $$ if for each $$\epsilon > 0\ $$, there is a measurable subset $$E_\epsilon \subseteq D$$ such that $$\mu(E_\epsilon) < \epsilon\ $$ and $$f_n\ $$ converges uniformly to $$f\ $$ on $$D - E_\epsilon\ $$.

Relations to Other Modes of Convergence
Uniform convergence a.e. is weaker than uniform convergence.

Uniform convergence a.e. implies convergence a.e. (proof here). A partial converse to this result is given by Egorov's Theorem.

Uniform convergence a.e. also implies convergence in measure.