Pointwise Limit of Measurable Functions is Measurable

Theorem

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

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

Then the pointwise limit $\ds \lim_{n \mathop \to \infty} f_n: X \to \overline \R$ is also $\Sigma$-measurable.

Proof

From Convergence of Limsup and Liminf, it follows that:

$\ds \lim_{n \mathop \to \infty} f_n = \limsup_{n \mathop \to \infty} f_n$

We have Pointwise Upper Limit of Measurable Functions is Measurable.

Hence the result.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.9$