Definition:Convergence in Measure

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Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

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


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

$\ds \forall \epsilon \in \R_{>0}: \lim_{n \mathop \to \infty} \map \mu {\set {x \in D : \size {\map {f_n} x - \map f x} \ge \epsilon} } = 0$

for all $D \in \Sigma$ with $\map \mu D < + \infty$.


To express that $f_n$ converges to $f$ in measure one writes $f_n \stackrel \mu \longrightarrow f$ or $\ds \underset {n \mathop \to \infty} {\mu \, \text - \lim \,} f_n = f$.


Technical Note

The expressions:

$f_n \stackrel \mu \longrightarrow f$
$\ds \underset {n \mathop \to \infty} {\mu \, \text - \lim \,} f_n = f$

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

f_n \stackrel \mu \longrightarrow f
\ds \underset {n \mathop \to \infty} {\mu \, \text - \lim \,} f_n = f


Sources