Definition:Convergence in Measure

Given a measure space $$(X, \Sigma, \mu)\ $$ and a sequence of $\Sigma$-measurable functions $$f_n:D\to\R$$ for $$D\in\Sigma$$, $$f_n\ $$ is said to converge in measure to a function $$f\ $$ on $$D\ $$ if
 * $$\lim_{n\to\infty}\mu(\{x\in D : |f_n(x) - f(x)| \geq \epsilon\}) = 0$$

for each $$\epsilon > 0\ $$. We also write $$f_n \stackrel{\mu}{\to} f$$ to express this property.