Convergence In Measure
Why
We want a form of the dominated convergence theorem in terms of convergence in measure.1
Definition
A sequence of real-valued measurable functions converges in measure to a real-valued measurable limit function if, for every positive number, the measures of the set where the function deviates from the limit function by more than the positive number converges to zero.
Notation
Suppose $\seq{f}$ is a sequence of real-valued
measurable funtions on a measure space $(X,
\mathcal{A} , \mu )$ and $f: X \to \R $
measurable.
If $f_n$ converges in measure to $f$ we write:
$f_n \goesto f$ in measure, read aloud as
“$f_n$ goes to $f$ in measure.”
This notation is an abbreviation of the
following relation
\[
\lim_{n \to \infty} \mu (\Set*{x \in X}{\abs{f_n(x) - f(x)} >
\epsilon }) = 0 \quad \text{ for all } \epsilon > 0
\]
- Future editions will expand. ↩︎