Prove: If $f \in L^p(\mathbb{R}^n)$, then $f$ is locally integrable.

Question

I have to prove the following statement, but I am unsure of my solution:

If $f \in L^p(\mathbb{R}^n)$, then $f$ is locally integrable for $p \geq 1$.

The definition of $f$ locally integrable is that $f$ is integrable on each compact set $K \subseteq \mathbb{R}^n$. In other words

$$\int_K|f| d \mu < \infty$$

Since $f \in L^p(\mathbb{R}^n)$,

$$||f||_p=\left| \int_{\mathbb{R}^n} |f|^p d\mu \right|^{1/p} < \infty \Rightarrow\int_{\mathbb{R}^n} |f|^p d\mu < \infty$$

$|f|^p$ is integrable iff $|f|^p\neq \infty$ almost everywhere. Therefore, $|f| \neq \infty$ almost everywhere, and

$$\int_{\mathbb{R}^n} |f| d\mu < \infty$$

Finally, for all $K \subset \mathbb{R}^n \to \mu(K) \leq \mu(\mathbb{R}^n)$ and

$$\int_K|f| d\mu \leq \int_{\mathbb{R}^n} |f| d\mu< \infty$$

I think my solution is wrong since I never used the compactness of $K$.

Answer 1

It is not correct.

First of all, it is recommended to write $|f|<\infty$ a.e. instead of $|f|\ne\infty$ since $0\leq|f|\leq\infty$.

And you can write $\displaystyle\int_{K}|f|d\mu\leq\int_{\mathbb{R}^{n}}|f|d\mu$ but this has nothing to do with $\mu(K)\leq\mu(\mathbb{R}^{n})$, the proper reasoning is just $|f|\chi_{K}\leq|f|$.

But it is not known whether $\displaystyle\int_{\mathbb{R}^{n}}|f|d\mu<\infty$. So it could end up with $\displaystyle\int_{K}|f|d\mu\leq\int_{\mathbb{R}^{n}}|f|d\mu=\infty$, so it is not conclusive to prove in this way.

You need Holder:

\begin{align*} \int_{K}|f|=\int\chi_{K}|f|\leq\|f\|_{L^{p}}\|\chi_{K}\|_{L^{q}} \end{align*} for $1/p+1/q=1$.

Answer 2

$\int |f| d\mu <\infty$ is false. You canot say that a function is integrable simply because it is finite almost everywhere.

Assume $p >1$ since the result is trivial for $p=1$. You have to use the fact that compact sets have finite measure. By Holder's inequlaity $\int_K |f| \leq (\int |f|^{p})^{1/p} (\mu (K))^{1/q}$ where $q =\frac p {p-1}$.

Answer 3

You can prove something stronger without Holder.

if X is a finite measure space and f is in $L^p(X)$ ($1\le p$).

Then f is in $L^q(X)$ for $1\le q\le p$ .

Let A be the set where $|f| \le 1$, let B be the set where $|f| \gt 1$. $|f|^q$ is integrable on A, because $|f|^q$ is bounded and the measure space is finite. $|f|^q$ is integrable on B, because in B, $|f|^q \le |f|^p$.