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 $f$ :


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