Vitali convergence theorem

Let ${\displaystyle (f_{n})_{n\in \mathbb {N} }\subseteq L^{p}(X,\tau ,\mu ),f\in L^{p}(X,\tau ,\mu )}$, with ${\displaystyle 1\leq p<\infty }$. Then, ${\displaystyle f_{n}\to f}$ in ${\displaystyle L^{p}}$ if and only if we have

• (i) ${\displaystyle f_{n}}$ converge in measure to ${\displaystyle f}$.
• (ii) For every ${\displaystyle \varepsilon >0}$ there exists a measurable set ${\displaystyle E_{\varepsilon }}$ with ${\displaystyle \mu (E_{\varepsilon })<\infty }$ such that for every ${\displaystyle G\in \tau }$ disjoint from ${\displaystyle E_{\varepsilon }}$ we have, for every ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \int _{G}|f_{n}|^{p}\,d\mu <\varepsilon ^{p}}$

• (iii) For every ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle \delta (\varepsilon )>0}$ such that, if ${\displaystyle E\in \tau }$ and ${\displaystyle \mu (E)<\delta (\varepsilon )}$ then, for every ${\displaystyle n\in \mathbb {N} }$ we have

${\displaystyle \int _{E}|f_{n}|^{p}\,d\mu <\varepsilon ^{p}}$

Remark: If ${\displaystyle \mu (X)}$ is finite, then the second condition is trivially true (just pick a subset that covers all but a sufficiently small portion of the whole range). Also, (i) and (iii) implies the uniform integrability of ${\displaystyle (|f_{n}|^{p})_{n\in \mathbb {N} }}$, and the uniform integrability of ${\displaystyle (|f_{n}|^{p})_{n\in \mathbb {N} }}$ implies (iii). [1]

For proving statement 1, we use Fatou's lemma: ${\displaystyle \int _{X}|f|\,d\mu \leq \liminf _{n\to \infty }\int _{X}|f_{n}|\,d\mu }$

• Using uniform integrability there exists ${\displaystyle \delta >0}$ such that we have ${\displaystyle \int _{E}|f_{n}|\,d\mu <1}$ for every set ${\displaystyle E}$ with ${\displaystyle \mu (E)<\delta }$
• By Egorov's theorem, ${\displaystyle {f_{n}}}$ converges uniformly on the set ${\displaystyle E^{C}}$. ${\displaystyle \int _{E^{C}}|f_{n}-f_{p}|\,d\mu <1}$ for a large ${\displaystyle p}$ and ${\displaystyle \forall n>p}$. Using triangle inequality, ${\displaystyle \int _{E^{C}}|f_{n}|\,d\mu \leq \int _{E^{C}}|f_{p}|\,d\mu +1=M}$
• Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.

For statement 2, use ${\displaystyle \int _{X}|f-f_{n}|\,d\mu \leq \int _{E}|f|\,d\mu +\int _{E}|f_{n}|\,d\mu +\int _{E^{C}}|f-f_{n}|\,d\mu }$, where ${\displaystyle E\in {\mathcal {F}}}$ and ${\displaystyle \mu (E)<\delta }$.

• The terms in the RHS are bounded respectively using Statement 1, uniform integrability of ${\displaystyle f_{n}}$ and Egorov's theorem for all ${\displaystyle n>N}$.

Let ${\displaystyle (X,{\mathcal {F}},\mu )}$ be a positive measure space. If

1. ${\displaystyle \mu (X)<\infty }$,
2. ${\displaystyle f_{n}\in {\mathcal {L}}^{1}(\mu )}$ and
3. ${\displaystyle \lim _{n\to \infty }\int _{E}f_{n}\,d\mu }$ exists for every ${\displaystyle E\in {\mathcal {F}}}$

then ${\displaystyle \{f_{n}\}}$ is uniformly integrable.[2]

• Modern methods in the calculus of variations. 2007. ISBN 9780387357843.
• Folland, Gerald B. (1999). Real analysis. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. pp. xvi+386. ISBN 0-471-31716-0. MR1681462
• Rosenthal, Jeffrey S. (2006). A first look at rigorous probability theory (Second ed.). Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. pp. xvi+219. ISBN 978-981-270-371-2. MR2279622

• Vitali convergence theorem at PlanetMath.