almost-everywhere convergence

In Lebesgue theory a handful of bad points never matters, as long as the bad set has measure zero. Almost-everywhere convergence relaxes ordinary pointwise convergence by forgiving failure on such a negligible set: the sequence converges at every point except possibly on a set you cannot even measure any positive size for. Since integrals are blind to measure-zero sets, this is the natural notion of convergence to feed the integral.

Precisely, a sequence of functions f_n converges almost everywhere (a.e.) to f if there is a set N with μ(N) = 0 such that f_n(x) converges to f(x) for every x outside N. The hypotheses in the monotone convergence theorem, Fatou's lemma, and dominated convergence are all stated with a.e. convergence, precisely because the integral ignores what happens on N.

Be careful what a.e. convergence does and does not give. It does not imply convergence in L^1 (the tall spikes converge to 0 a.e. but their integrals stay at 1) nor convergence in measure in general for arbitrary spaces, and the limit f is only determined up to a measure-zero set. Conversely, convergence in measure does not imply a.e. convergence, though it forces a subsequence that does converge a.e.