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

for each $$\epsilon > 0\ $$.

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