Convex Real Function is Measurable
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Theorem
Let $f : \R \to \R$ be a convex real function.
Then $f$ is measurable.
Proof 1
From Convex Real Function is Continuous, $f$ is continuous.
From Continuous Mapping is Measurable, $f$ is measurable.
$\blacksquare$
Proof 2
From Convex Real Function is Pointwise Supremum of Affine Functions: Corollary, there exists a countable set $\SS \subseteq \R^2$ such that:
- $\ds \map f x = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a x + b}$
for each $x \in \R$.
From Linear Function is Continuous, the map $x \mapsto a x + b$ is continuous for each $\tuple {a, b} \in \SS$.
From Continuous Mapping is Measurable, the map $x \mapsto a x + b$ is measurable.
Since $\SS$ is countable, we can apply Pointwise Supremum of Measurable Functions is Measurable to obtain that $f$ is measurable.
$\blacksquare$