Let $(X, \Sigma, \mu)$ be a measure space and $f$ be a measurable function. For each $1\le p<+\infty$, setting $$\|f\|_{p}:=\left(\int\limits_{X}|f|^{p}\, \mathrm{d}\mu\right)^{\frac{1}{p}} \text{(not necessary finite)}.$$ The question is that $$\|f\|_{p}\to \|f\|_{p_{0}}\, \text{ as }\, p\to p_{0}?$$
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