Definition:Almost Uniform Convergence

Given a measure space $$(X, \Sigma, \mu)\ $$ and a sequence of 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 $$D - E_\epsilon\ $$.