modes of convergence of sequences of measurable functions
Let be a measure space, be measurable functions
for every positive integer , and be a measurable function. The following are modes of convergence of :
- •
converges almost everywhere to if
- •
converges almost uniformly to if, for every , there exists with and converges uniformly to on
- •
converges in measure to if, for every , there exists a positive integer such that, for every positive integer , .
- •
If, in , and each are also Lebesgue integrable
, converges in to if .
A lot of theorems in real analysis (http://planetmath.org/BibliographyForRealAnalysis) deal with these modes of convergence. For example, Fatou’s lemma, Lebesgue’s monotone convergence theorem, and Lebesgue’s dominated convergence theorem give conditions on sequences
of measurable functions that converge almost everywhere under which they also converge in . Also, Egorov’s theorem that, if , then convergence almost everywhere implies almost uniform convergence
.