Survival Function is Decreasing

Theorem

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

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

Let $F_f$ be the survival function of $f$.

Then $F_f$ is a decreasing function.

Proof

Let $\alpha, \beta \in \hointr 0 \infty$ with $\alpha \le \beta$.

We show:

$\map {F_f} \beta \le \map {F_f} \alpha$

We first show:

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

Let $x \in X$ have $\size {\map f x} \ge \beta$.

Then, since $\beta \ge \alpha$ we have $\size {\map f x} \ge \alpha$.

So by the definition of set inclusion, we have:

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

So, by Measure is Monotone we have:

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

So by the definition of the survival function we have:

$\map {F_f} \beta \le \map {F_f} \alpha$

So $F_f$ is a decreasing function.

$\blacksquare$

Sources

• 2014: Loukas Grafakos: Classical Fourier Analysis (3rd ed.) ... (previous) ... (next): $1.1.1$: The Distribution Function