Definition:Convergence in Measure

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

Let $D \in \Sigma$.

Let $f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions.

Then $f_n$ is said to converge in measure to a function $f$ on $D$ if:


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

We also write $f_n \stackrel{\mu}{\to} f$ to express this property.