For $p\geq1$, $f\in L^p(\mathbb{R}^n)$ and $\lambda>0$, does $\{x\in\mathbb{R}^n:|f(x)|>\lambda\}$ have finite measure?

Question

I have very little background in measure theory but this has come up as part of a proof I'm trying to construct, so would appreciate some help (sorry if it's obvious).

Suppose $f\in L^p(\mathbb{R}^n)$ where $p\geq1$, and let $\lambda>0$ be some real number. Then is it true that the Lebesgue measure of the set $\{x\in\mathbb{R}^n:|f(x)|>\lambda\}$ is finite? My intuition tells me yes, since in order for the integral \begin{equation} \int_{\mathbb{R}^n} |f(x)|^p\,\mathrm{d}x \end{equation} to be finite it seems that $f$ must decay on some 'significant' portion of the domain $\mathbb{R}^n$, and consequently only take absolute values greater than $\lambda$ on some finite (compact?) subset of $\mathbb{R}^n$. How would one go about proving this, or alternative provide some counterexample?

Thanks in advance

Answer 1

Yes, for if not we see that with $S_\lambda = \{x : |f|>\lambda\}$

$$\infty = \mu(S_\lambda)\cdot \lambda^p = \int_{S_\lambda}\lambda^p\le \int_{S_\lambda}|f|^p\le \int_{\Omega}|f|^p<\infty$$

Answer 2

$$ |\{|f|>\lambda\}|=\int_{|f|/\lambda>1}dx\le\int_{\mathbb{R}^n}\frac{|f|^p}{\lambda^p}\,dx=\frac{\|f\|_p^p}{\lambda^p}. $$