Proofs of properties of a measureable and Lebesgue integrable function

Question

Could I get some help showing the following properties to be true:

a) $f: X \to [0,\infty) $ is measurable and $\int f d\mu < \infty$

$\forall a > 0$, let $X_a = \{x \in X :f(x) >a\}$, show $\mu(X_a) \leq \frac{1}{a} \int f d\mu$.

b) Suppose $g:\mathbb{R} \to [0,\infty)$ vanishes outside a bounded interval and $\int g^2 d\lambda < \infty$

Show that $\int g d\lambda < \infty$

Thank you for your help

Answer 1

a) $\int f d\mu = \int_{X_a} f d\mu + \int_{(X_a)^c} f d\mu $. But $\int_{X_a} f d\mu > a\mu(X_a)$ and since $f$ is positive $\int_{(X_a)^c} f d\mu>0$. Therefore $\int f d\mu > a\mu(X_a)$ which gives what you want.

b)You have the Cauchy Schwarz inequality which says for every $f$, $g$ in $L^2(\mu)$

$$\int |fg| < \left(\int |f|^2 \right)^{1/2}\left(\int |g|^2 \right)^{1/2}$$

Placing $f=1$ you get that

$$\int |g| < (\int |g|^2)^{1/2}$$

In your case since $g>0$ you can replace $|g|$ by $g$.

Answer 2

1. For all $a > 0$,

\begin{equation} \mu(X_a) = \int_X 1_{X_a}\, d\mu = \frac{1}{a}\int_X a1_{X_a}\, d\mu \le \frac{1}{a}\int_X f1_{X_a}\, d\mu \le \frac{1}{a} \int_X f\, d\mu. \end{equation}

2. Suppose $g(x) = 0$ for all $x\notin [a,b]$. By the Cauchy-Schwarz inequality,

\begin{equation} \int_{\Bbb R} g\, d\lambda = \int_{\Bbb R} g1_{[a,b]}\, d\lambda \le \left(\int_{\Bbb R} 1_{[a,b]}^2\, d\lambda\right)^{1/2} \left(\int_{\Bbb R} g^2\, d\lambda\right)^{1/2} = (b - a)^{1/2}\left(\int_{\Bbb R} g^2\, d\lambda\right)^{1/2} < \infty. \end{equation}