Pointwise Minimum of Measurable Functions is Measurable
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Theorem
Let $\struct {X, \Sigma}$ be a measurable space.
Let $f, g: X \to \overline \R$ be $\Sigma$-measurable functions.
Then the pointwise minimum $\min \set {f, g}: X \to \overline \R$ is also $\Sigma$-measurable.
Proof
For all $x \in X$ and $a \in \R$, we have by Min Operation Yields Infimum of Parameters that:
- $a \le \min \set {\map f x, \map g x}$
if and only if both $a \le \map f x$ and $a \le \map g x$.
That is, for all $a \in \R$:
- $\set {x \in X: \min \set {\map f x, \map g x} \ge a} = \set {x \in X: \map f x \ge a} \cap \set {x \in X: \map g x \ge a}$
By Characterization of Measurable Functions: $(1) \implies (4)$, the two sets on the RHS are elements of $\Sigma$, that is, measurable.
By Sigma-Algebra Closed under Intersection:
- $\set {x \in X: \min \set {\map f x, \map g x} \ge a} \in \Sigma$
Hence $\min \set {f, g}$ is measurable, by Characterization of Measurable Functions: $(4) \implies (1)$.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.10$