Vitali convergence theorem

Let $f_{1},f_{2},\ldots$ be $\mathbf{L}^{p}$ -integrable functions on some measure space, for $1\leq p<\infty$ .

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

i

the sequence $\{f_{n}\}$ converges to $f$ in measure;
ii

the functions $\{\lvert f_{n}\rvert^{p}\}$ are uniformly integrable; and
iii

for every $\epsilon>0$ , there exists a set $E$ of finite measure, such that $\int_{E^{\mathrm{c}}}\lvert f_{n}\rvert^{p}<\epsilon$ for all $n$ .

Remarks

This theorem can be used as a replacement for the more well-known dominated convergence theorem, when a dominating 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 , 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.

References

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.

Title	Vitali convergence theorem
Canonical name	VitaliConvergenceTheorem
Date of creation	2013-03-22 16:17:10
Last modified on	2013-03-22 16:17:10
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	9
Author	stevecheng (10074)
Entry type	Theorem
Classification	msc 28A20
Synonym	uniform-integrability convergence theorem
Related topic	ModesOfConvergenceOfSequencesOfMeasurableFunctions
Related topic	UniformlyIntegrable
Related topic	DominatedConvergenceTheorem