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 $F_{c f}$ be the survival function of the pointwise scalar multiple $c f$.

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

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

Then, we have: