Definition:Convergence in Mean

Theorem

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

Let $f : X \to \R$ be a $\mu$-integrable function.

For each $n \in \N$, let $f_n : X \to \R$ be a $\mu$-integrable function.

We say that the sequence $\sequence {f_n}_{n \mathop \in \N}$ converges in mean to $f$ if and only if:

$\ds \lim_{n \mathop \to \infty} \int \size {f_n - f} \rd \mu = 0$

Also see

• Results about convergence in mean can be found here.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $3.1$