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

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

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

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

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

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

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 $\lambda \to\infty$, for if $|f|^p$ is integrable, it must be $<\infty$ almost everywhere.

3. Thanks! That was really dumb of me.