Definition:Convergence in Measure

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Definition

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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


Then $f_n$ is said to converge in measure to a measurable function $f: X \to \R$ iff:

$\displaystyle \forall \epsilon > 0: \lim_{n \to \infty} \mu \left({ \left\{ {x \in D : \left|{ f_n \left({ x }\right) - f \left({ x }\right) }\right| \ge \epsilon }\right\} }\right) = 0$

for all $D \in \Sigma$ with $\mu \left({D}\right) < + \infty$.


To express that $f_n$ converges to $f$ in measure one writes $f_n \stackrel{\mu}{\longrightarrow} f$ or $\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f$.


Technical Note

The expressions:

$f_n \stackrel{\mu}{\longrightarrow} f$
$\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f$

are produced by the following (intricate) $\LaTeX$ code:

f_n \stackrel{\mu}{\longrightarrow} f
\displaystyle \operatorname{\mu-\!\lim\,} \limits_{n \to \infty} f_n = f


Sources