Convergence in measure

A convergence in measure is the following notion of convergence for a sequence $(f_n)$ of measurable functions on a measure space $(X,\mathcal{A},\mu)$ : we say that $f_n$ converges in measure to a measurable function $f$ if for every $\varepsilon>0$ ,

\mu\big(\{x\in X:\ |f_n(x)-f(x)|>\varepsilon\}\big)\to 0 \quad\text{as }n\to\infty.

This definition treats two functions as close whenever they differ by more than $\varepsilon$ only on a set of small measure. If $\mu(X)<\infty$ , then convergence almost everywhere implies convergence in measure, but the converse can fail.

Examples

On $([0,1],\mathcal{B},\lambda)$ with Lebesgue measure $\lambda$ , the functions $f_n=\mathbf{1}_{[0,1/n]}$ satisfy $f_n\to 0$ in measure because $\lambda(\{|f_n|>1/2\})=\lambda([0,1/n])=1/n\to 0$ .
On $(\mathbb{R},\mathcal{B},\lambda)$ , the functions $f_n=\mathbf{1}_{[n,n+1]}$ do not converge in measure to $0$ because for $\varepsilon=\tfrac12$ one has $\lambda(\{|f_n|>\varepsilon\})=\lambda([n,n+1])=1$ for all $n$ .