# 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.

$\blacksquare$