Vitali's Theorem

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

Let $\sequence {f_n}_{n \mathop \in \N}: f_n: X \to \R$ be a sequence of $p$-integrable functions.

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

Suppose that $\ds \operatorname {\mu-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$, that is $f_n$ converges in measure to $f$.

Then the following are equivalent:


 * $(1): \quad \ds \lim_{n \mathop \to \infty} \norm {f_n - f}_p = 0$, where $\norm {\,\cdot\,}_p$ is the $p$-seminorm (that is $\ds \operatorname {\LL^{\textit p}-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$)
 * $(2): \quad \paren {\size {f_n}^p}_{n \mathop \in \N}$ is a uniformly integrable collection
 * $(3): \quad \ds \lim_{n \mathop \to \infty} \int \size {f_n}^p \rd \mu = \int \size f^p \rd \mu$

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


 * $(1'): \quad \sequence {f_n}_{n \mathop \in \N}$ converges in $\LL^p$
 * $(3'): \quad \ds \lim_{n \mathop \to \infty} \int \size {f_n}^p \rd \mu$ exists in $\R$