Definition:Convergence in Measure

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 $f: X \to \R$ :


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

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

Technical Note
The expressions:


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

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

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