Category:Scheffé's Lemma

This category contains pages concerning Scheffé's Lemma:

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

Let $f_n$ be a sequence of $\mu$-integrable functions that converge almost everywhere to another $\mu$-integrable function $f$.

Then $f_n$ converges to $f$ in $L^1$ if and only if $\ds \int_X \size {f_n} \rd \mu$ converges to $\ds \int_X \size f \rd \mu$.