Vitali convergence theorem

Let be $\Le^p$ -integrable functions on some measure space, for $1 \leq p < \infty$ .

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

1. the sequence $\{ f_n \}$ converges to $f$ in measure;
2. the functions $\{ \abs{f_n}^p \}$ are uniformly integrable; and
3. for every $\epsilon > 0$ , there exists a set $E$ of finite measure, such that $\int_{E^\mathrm{c}} \abs{f_n}^p < \epsilon$ for all $n$ .

Remarks

This theorem can be used as a replacement for the more well-known dominated convergence theorem, when a dominating factor cannot be found for the functions $f_n$ to be integrated. (If this theorem is known, the dominated convergence theorem can 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 considerations in the general case to the case of a finite measure space.

In probability theory, 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.

Bibliography

1

Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.

2

Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.