\begin{definition}[Definition:Convergence in Measure]
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 $f: X \to \R$ {{iff}}:
:$\ds \forall \epsilon \in \R_{>0}: \lim_{n \mathop \to \infty} \map \mu {\set {x \in D : \size {\map {f_n} x - \map f x} \ge \epsilon} } = 0$
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 $\ds \underset {n \mathop \to \infty} {\mu \, \text - \lim \,} f_n = f$.
\end{definition}