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
Also see
- Results about convergence in measure can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $16.2$