Editing Dominated convergence theorem (section)

==Dominated convergence in ''L''<sup>''p''</sup>-spaces (corollary)==
Let <math>(\Omega,\mathcal{A},\mu)</math> be a [[measure space]], {{nowrap|<math> 1\leq p<\infty</math>}} a real number and <math>(f_n)</math> a sequence of <math>\mathcal{A}</math>-measurable functions <math>f_n:\Omega\to\Complex\cup\{\infty\}</math>.

Assume the sequence <math>(f_n)</math> converges <math>\mu</math>-almost everywhere to an <math>\mathcal{A}</math>-measurable function <math>f</math>, and is dominated by a <math>g \in L^p</math> (cf. [[Lp space]]), i.e., for every natural number <math>n</math> we have: <math>|f_n|\leq g</math>, μ-almost everywhere.

Then all <math>f_n</math> as well as <math>f</math> are in <math>L^p</math> and the sequence <math>(f_n)</math> converges to <math>f</math> in [[Lp-space|the sense of <math>L^p</math>]], i.e.:

:<math>\lim_{n \to \infty}\|f_n-f\|_p =\lim_{n \to \infty}\left(\int_\Omega |f_n-f|^p \,d\mu\right)^{\frac{1}{p}} = 0.</math>

Idea of the proof: Apply the original theorem to the function sequence <math>h_n = |f_n-f|^p</math> with the dominating function <math>(2g)^p</math>.