Survival Function of Pointwise Scalar Multiple

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 $f$.

Let $c \in \R \setminus \set 0$.

Then:

$\ds \map {F_{c f} } \alpha = \map {F_f} {\frac \alpha {\size c} }$ for all $\alpha \in \hointr 0 \infty$.

Let $\alpha \in \hointr 0 \infty$.

Then, we have:

$\ds \map {F_{c f} } \alpha$

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds \map {F_f} {\frac \alpha {\size c} }$

$\blacksquare$