Survival Function is Well-Defined

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Theorem

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

Let $f: X \to \overline \R$ be a $\Sigma$-measurable function.

Then the survival function $F_f$ is well-defined.

Proof

From Absolute Value of Measurable Function is Measurable:

$\size f$ is $\Sigma$-measurable.

Then from Characterization of Measurable Functions, we have:

$\set {x \in X : \size {\map f x} \ge \alpha} \in \Sigma$ for all $\alpha \in \hointr 0 \infty$.

So:

$\map \mu {\set {x \in X : \size {\map f x} \ge \alpha} }$

is well-understood, and the survival function $F_f$ is well-defined.

$\blacksquare$