Pointwise Lower 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 lower limit:

$\ds \liminf_{n \mathop \to \infty} f_n: X \to \overline \R$

is also $\Sigma$-measurable.

Proof

By definition of limit inferior, we have:

$\ds \liminf_{n \mathop \to \infty} f_n = \sup_{m \mathop \in \N} \ \inf_{n \mathop \ge m} f_n$

The result follows from combining:

Pointwise Infimum of Measurable Functions is Measurable

Pointwise Supremum of Measurable Functions is Measurable

$\blacksquare$

Sources

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