Convergence in measure

Summary

A sequence $(f_n)$ of measurable functions converges in measure to $f$ if bad sets shrink. For each $\varepsilon > 0$, the set where $f_n$ differs from $f$ by at least $\varepsilon$ must have measure tending to $0$. It is weaker than uniform convergence and than convergence in $L^p$. In probability theory it is called convergence in probability.