Survival Function is Decreasing
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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.
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