Vitali's Theorem

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space, and let $p \in \R, p \ge 1$.

Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence of $p$-integrable functions.

Also, let $f: X \to \R$ be a measurable function.


Suppose that $\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f$, i.e. $f_n$ converges in measure to $f$.


Then the following are equivalent:

$(1): \quad \displaystyle \lim_{n \to \infty} \left\Vert{f_n - f}\right\Vert_p = 0$, where $\left\Vert{\cdot}\right\Vert_p$ is the $p$-seminorm (i.e. $\displaystyle \operatorname{\mathcal L^{\textit p}-\!\lim\,} \limits_{n \to \infty} f_n = f$)
$(2): \left({\left\vert{f_n}\right\vert^p}\right)_{n \in \N}$ is a uniformly integrable collection
$(3): \displaystyle \lim_{n \to \infty} \int \left\vert{f_n}\right\vert^p \, \mathrm d \mu = \int \left\vert{f}\right\vert^p \, \mathrm d \mu$

If $\left({X, \Sigma, \mu}\right)$ is not $\sigma$-finite, $(1)$ and $(3)$ must be replaced by, respectively:

$(1'): \quad \left({f_n}\right)_{n \in \N}$ converges in $\mathcal L^p$
$(3'): \quad \displaystyle \lim_{n \to \infty} \int \left\vert{f_n}\right\vert^p \, \mathrm d \mu$ exists in $\R$


Proof


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