Vitali convergence theorem

Let $f_1,f_2,\mathrm{\dots }$ be ${𝐋}^p$-integrable functions on some measure space, for .

The sequence $\{f_n\}$ converges in ${𝐋}^p$ to a measurable function $f$if and and only if

1. the sequence $\{f_n\}$ converges to $f$ in measure;

2. the functions $\{{|f_n|}^p\}$ are uniformly integrable; and

3. for every $ϵ>0$, there exists a set $E$of finite measure, such that for all $n$.

Remarks

This theorem can be used as a replacement for the morewell-known dominated convergence theorem, when adominating cannot be found for the functions$f_n$ to be integrated.(If this theorem is known, the dominated convergence theoremcan be derived as a special case.)

In a finite measure space, condition (iii) is trivial.In fact, condition (iii) is the tool used to reduce considerationsin the general case to the case of a finite measure space.

In probability , the definition of “uniform integrability”is slightly different from its definition in general measure theory;either definition may be used in the statement of this theorem.