Category:Definitions/Almost Uniform Convergence

This category contains definitions related to Almost Uniform Convergence.
Related results can be found in Category:Almost Uniform Convergence.

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

Let $D \in \Sigma$.

Let $\sequence {f_n}_{n \mathop \in \N}, f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions.

Then $\sequence {f_n}_{n \mathop \in \N}$ is said to converge almost uniformly on $D$ if and only if:

For all $\epsilon > 0$, there is a measurable subset $E_\epsilon \subseteq D$ of $D$ such that:

$(1): \quad \map \mu {E_\epsilon} < \epsilon$

$(2): \quad \sequence {f_n}_{n \mathop \in \N}$ converges uniformly to $f$ on $D \setminus E_\epsilon$.