Bound on Survival Function of Pointwise Sum

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Theorem

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

Let $f, g : X \to \overline \R$ be $\Sigma$-measurable functions such that the pointwise sum $f + g$ is well-defined.

Let $F_{f + g}$ be the survival function of the pointwise scalar multiple $f + g$.

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 $\map f x + \map g x = \infty$ or $\map f x + \map g x = -\infty$.

In the former case, we have $\map f x = \infty$ or $\map g x = \infty$.

In the latter case, we have $\map f x = -\infty$ or $\map g x = -\infty$.

So in either case we have $\size {\map f x} = \infty$ or $\size {\map g x} = \infty$.

So we either have $\size {\map f x} \ge \alpha$ or $\size {\map g x} \ge \beta$.

So:

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

in this case.

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 from the Triangle Inequality, we have:

$\size {\map f x + \map g x} \le \size {\map f x} + \size {\map g x} < \alpha + \beta$

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:

$\set {x \in X : \size {\map f x + \map g x} \ge \alpha + \beta} \subseteq \set {x \in X : \size {\map f x} \ge \alpha} \cup \set {x \in X : \size {\map g x} \ge \beta}$ for all $\alpha, \beta \in \hointr 0 \infty$.

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

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