# Thread: Limit of the Measure of a Set

1. ## Limit of the Measure of a Set

Hello. I am working on the following problem:

Let $\displaystyle (X,\mathcal{A},\mu)$ be a measure space and let $\displaystyle f$ be an extended real-valued $\displaystyle \mathcal{A}$-measurable function on $\displaystyle X$ such that $\displaystyle \int_X |f|^p\text{ d}\mu<\infty$ for some $p\in (0,\infty)$. Show that $\displaystyle \lim_{\lambda\to \infty} \lambda^p \mu\{X:|f|\geq \lambda\}=0$.

Now, I let $\displaystyle X_\lambda=\{X:|f|\geq\lambda\}$ and tried to consider the inequalities

$\displaystyle \displaystyle \int_X |f|^p\text{ d}\mu\geq \int_{X_\lambda} |f|^p\text{ d}\mu\geq \int_{X_\lambda} \lambda^p\text{ d}\mu=\lambda^p \mu(X_\lambda)$

The expression on the left side is finite, but as far as I can tell, this doesn't give much information about the expression on the right. At this point, I'm at a loss on this problem, so I would appreciate if anybody had some suggestions.

Thanks.

2. Originally Posted by roninpro
Hello. I am working on the following problem:

Let $\displaystyle (X,\mathcal{A},\mu)$ be a measure space and let $\displaystyle f$ be an extended real-valued $\displaystyle \mathcal{A}$-measurable function on $\displaystyle X$ such that $\displaystyle \int_X |f|^p\text{ d}\mu<\infty$ for some $p\in (0,\infty)$. Show that $\displaystyle \lim_{\lambda\to \infty} \lambda^p \mu\{X:|f|\geq \lambda\}=0$.

Now, I let $\displaystyle X_\lambda=\{X:|f|\geq\lambda\}$ and tried to consider the inequalities

$\displaystyle \displaystyle \int_X |f|^p\text{ d}\mu\geq \int_{X_\lambda} |f|^p\text{ d}\mu\geq \int_{X_\lambda} \lambda^p\text{ d}\mu=\lambda^p \mu(X_\lambda)$

The expression on the left side is finite, but as far as I can tell, this doesn't give much information about the expression on the right. At this point, I'm at a loss on this problem, so I would appreciate if anybody had some suggestions.

Thanks.
Surely, the second integral from the left goes to 0 as $\displaystyle \lambda \to\infty$, for if $\displaystyle |f|^p$ is integrable, it must be $\displaystyle <\infty$ almost everywhere.

3. Thanks! That was really dumb of me.