Could someone help me prove that given a finite measure space (X,M,σ) and a measurable function f:X→R in L∞ and some Lq, limp→∞‖?

I don’t know where to start.

**Answer**

Fix \delta>0 and let S_\delta:=\{x,|f(x)|\geqslant \lVert f\rVert_\infty-\delta\} for \delta<\lVert f\rVert_\infty. We have

\lVert f\rVert_p\geqslant \left(\int_{S_\delta}(\lVert f\rVert_\infty-\delta)^pd\mu\right)^{1/p}=(\lVert f\rVert_\infty-\delta)\mu(S_\delta)^{1/p},

since \mu(S_\delta) is finite and positive.

This gives

\liminf_{p\to +\infty}\lVert f\rVert_p\geqslant\lVert f\rVert_\infty.

As |f(x)|\leqslant\lVert f\rVert_\infty for almost every x, we have for p>q,

\lVert f\rVert_p\leqslant\left(\int_X|f(x)|^{p-q}|f(x)|^qd\mu\right)^{1/p}\leqslant \lVert f\rVert_\infty^{\frac{p-q}p}\lVert f\rVert_q^{q/p},

giving the reverse inequality.

**Attribution***Source : Link , Question Author : Parakee , Answer Author : Community*