Bound on Survival Function of Pointwise Product
Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g : X \to \overline \R$ be $\Sigma$-measurable functions.
Let $F_{f g}$ be the survival function of the pointwise product $f g$.
Let $F_f$ and $F_g$ be the survival functions of $f$ and $g$ respectively.
Then:
- $\ds \map {F_{f g} } {\alpha \beta} \le \map {F_f} \alpha + \map {F_g} \beta$ for all $\alpha, \beta \in \hointr 0 \infty$.
Proof
Let $\alpha, \beta \in \hointr 0 \infty$.
We show that:
- $\set {x \in X : \size {\map f x \map g x} \ge \alpha \beta} \subseteq \set {x \in X : \size {\map f x} \ge \alpha} \cup \set {x \in X : \size {\map g x} \ge \beta}$
Let $x \in X$ be such that $\size {\map f x \map g x} \ge \alpha \beta$.
If $\size {\map f x \map g x} = \infty$, we have $\size {\map f x} = \infty$ or $\size {\map g x}$.
Then we either have $\size {\map f x} \ge \alpha$ or $\size {\map g x} \ge \beta$, so that:
- $x \in \set {x \in X : \size {\map f x} \ge \alpha} \cup \set {x \in X : \size {\map g x} \ge \beta}$.
Now suppose that $\size {\map f x \map g x} < \infty$.
Suppose that $\size {\map f x} < \alpha$ and $\size {\map g x} < \beta$, then $\size {\map f x \map g x} = \size {\map f x} \size {\map g x} < \alpha \beta$
contradicting that:
- $\size {\map f x \map g x} \ge \alpha \beta$
So we must have $\size {\map f x} \ge \alpha$ or $\size {\map g x} \ge \beta$.
So we have:
- $x \in \set {x \in X : \size {\map f x} \ge \alpha} \cup \set {x \in X : \size {\map g x} \ge \beta}$.
again in this case.
So:
- $\set {x \in X : \size {\map f x \map g x} \ge \alpha \beta} \subseteq \set {x \in X : \size {\map f x} \ge \alpha} \cup \set {x \in X : \size {\map g x} \ge \beta}$
by the definition of set inclusion.
Then we have:
\(\ds \map {F_{f g} } {\alpha \beta}\) | \(=\) | \(\ds \map \mu {\set {x \in X : \size {\map f x \map g x} \ge \alpha \beta} }\) | Definition of Survival Function | |||||||||||
\(\ds \) | \(\le\) | \(\ds \map \mu {\set {x \in X : \size {\map f x} \ge \alpha} \cup \set {x \in X : \size {\map g x} \ge \beta} }\) | Measure is Monotone | |||||||||||
\(\ds \) | \(\le\) | \(\ds \map \mu {\set {x \in X : \size {\map f x} \ge \alpha} } + \map \mu {\set {x \in X : \size {\map g x} \ge \beta} }\) | Measure is Subadditive | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {F_f} \alpha + \map {F_g} \beta\) | Definition of Survival Function |
$\blacksquare$
Sources
- 2014: Loukas Grafakos: Classical Fourier Analysis (3rd ed.) ... (previous) ... (next): $1.1.1$: The Distribution Function