= Moment ratios converge to the essential supremum
{title2=$M_{n+1}/M_n\to\|f\|_\infty$}
For a bounded measurable function with positive essential supremum on a finite measure space, put $M_n=\int|f|^n$. The <Cauchy-Schwarz inequality> makes these moments log-convex, so successive ratios are nondecreasing. Their product satisfies $M_n/M_0\leq(M_{n+1}/M_n)^n$, while each ratio is at most the <essential supremum>. Positive-measure level sets arbitrarily close to that supremum make $\mu(E)^{-1/n}\|f\|_n$ converge to it. Squeezing then proves the ratio limit.
Back to article page
