Definition:Cauchy in Measure

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of real-valued $\Sigma$-measurable functions.

We say that $\sequence {f_n}_{n \mathop \in \N}$ is Cauchy in measure if and only if:

for each $\epsilon > 0$, there exists $N \in \N$ such that for all $n, m \ge N$ we have:

$\map \mu {\set {x \in X : \size {\map {f_n} x - \map {f_m} x} > \epsilon} } < \epsilon$

Sources

• 2014: Loukas Grafakos: Classical Fourier Analysis (3rd ed.) ... (previous) ... (next): $1.1.2$: Convergence in Measure