prove that $\lim_{p\to 0^+}\int_X |f|^p d\mu = \mu\Big(\{x\in X :f(x)\neq 0\} \Big)$

Question

Let $(X,\mathcal{M}, \mu)$ be a measure space with a finite measure $\mu$ and let $f$ be a measurable function such that $\int_X |f|^q d\mu <\infty$ for some $0 < q < \infty$. Show that $$\lim_{p\to 0^+}\int_X |f|^p d\mu = \mu\Big(\{x\in X :f(x)\neq 0\} \Big)$$

(I wrote my attempt but since it was based on a false assumption I deleted that, so wondered if someone can give a hint about this)

Answer 1

Notice that $|f|^p\leq 1_{\{|f|\leq 1\}}+|f|^q1_{\{|f|> 1\}}$ for $0<p\leq q$. The function on the right is integrable since $$ \int_X 1_{\{|f|\leq 1\}}+|f|^q1_{\{|f|> 1\}} \leq \mu(X)+\int_X |f|^q<\infty. $$ Now we can apply DCT since $|f|^p\leq 1_{\{|f|\leq 1\}}+|f|^q1_{\{|f|> 1\}}$. We have $$ \lim_{p\to 0^+}\int_X |f|^p=\int_X \lim_{p\to 0^+}|f|^p=\int_{\{|f|\neq 0\}}1=\mu(|f|\neq 0) $$ NOTE: Since $f\in L^q(X)$, $|f|<\infty$ almost everywhere, so in particular, $\lim_{p\to 0^+}|f|^p=0$ almost everywhere.

Answer 2

Consider any decreasing sequence $(p_{n})$ to zero. Write \begin{align*} \int|f|^{p_{n}}=\int_{0<|f|\leq 1}|f|^{p_{n}}+\int_{|f|>1}|f|^{p_{n}}. \end{align*} The second integral is monotone decreasing, and $\displaystyle\int_{|f|>1}|f|^{p_{1}}$ is integrable (we choose $p_{1}$ small enough such that $p_{1}<q$), so it converges to $\displaystyle\int_{|f|>1}1$.

The first integral is monotone increasing, and we apply Monotone Convergence Theorem to get the limit as $\displaystyle\int_{0<|f|\leq 1}1.$