Non-increasing Function of Measurable Sets

Question

I'm stuck trying to solve the following problem:

Let $f$ be a non-negative measurable function on a finite measure space $(X,\mathfrak{M},\mu)$ and let $\lambda(t)=\mu(\{x\in X~:~f(x)>t\})$. Prove that $\lambda$ is a non-increasing function of $t$ and is continuous from the right; i.e. $\lim_{h\searrow 0}\lambda(t+h)=\lambda(t)$.

So, to show that $\lambda$ is non-increasing, I believe we need to show that $\lambda(t_{1})\geq\lambda(t_{2})$ whenever $t_{1}<t_{2}$. I'm assuming that I need to use the fact that $(X,\mathfrak{M},\mu)$ is a finite measure space to show that $\lambda$ is a non-increasing function in $t$. Other than this, I have no ideas on how to tackle this problem. Any help is appreciated!

Answer 1

If $t_1< t_2,$ then $f(x)>t_2 $ implies that $t(x)>t_1,$ so you have $$ \{x\in X :f(x)>t_2\}\subseteq \{x\in X :f(x)>t_1\},$$ so since the measure of a measurable set is always greater than or equal to the measure of a measurable subset, we have $ \lambda(t_2)\le \lambda(t_1).$

For the continuity part, use the theorem about continuity of measure.

Answer 2

Here are some hints:

Suppose $t_1<t_2$. What can you say about the sets $\{x\in X:f(x)>t_1\}$ and $\{x\in X:f(x)>t_2\}$? Is one a subset of the other?

To show continuity from the right, it suffices to show that if $(h_n)$ is a decreasing sequence of positive real numbers such that $\lim_{n\to\infty}h_n=0$, then $\lim\lambda(t+h_n)=\lambda(t)$. Can you deduce anything about the sets $\{x\in X:f(x)>t+h_n\}$, $\{x\in X:f(x)>t\}$? Once you have that, use some sort of continuity of measures to deduce the right-continuity of $\lambda$ (this is where finiteness of $\mu$ comes into the picture).