The following is an old real analysis qual problem which I cannot solve.
Problem. Let $f\geq 0$ be a measurable function on $\mathbb{R}^{n}$. Suppose there exists $C>0$ such that for all Lebesgue measurable sets $D$ with positive Lebesgue measure $\left|D\right|$,
$$\int_{D}f^{1/\left|D\right|}dx\leq C \tag{1}$$
Show that $f$ vanishes outside a set of measure $C$, and that $f\leq 1$ a.e.
I believe I can show using Fatou's lemma that $\left|\left\{f>0\right\}\right|=0$, but I cannot show that $\left|\left\{f>0\right\}\right|\leq C$ or that $\left|\left\{f>1\right\}\right|=0$. Any suggestions would be appreciated.
This isn't hard; I was being obtuse. It is clear from the hypothesis that $\left|\left\{f=+\infty\right\}\right|=0$. Set $D_{j}:=B(0,j)\subset\mathbb{R}^{n}$, the ball of radius $j\in\mathbb{N}$ centered at the origin, and define a sequence $(f_{j})$ of nonnegative measurable functions by
$$f_{j}:=f^{1/\left|D_{j}\right|}\chi_{f>0}\chi_{D_{j}}$$
Since $f(x)^{1/\left|D_{j}\right|}\rightarrow 1$ for a.e. every $x$ such that $f(x)\neq 0$, we have by Fatou's lemma,
$$\left|\left\{f>0\right\}\right|=\int_{\mathbb{R}^{n}}\liminf_{j\rightarrow\infty}f_{j}dx\leq\liminf_{j\rightarrow\infty}\int_{\mathbb{R}^{n}}f_{j}=\liminf_{j\rightarrow\infty}\int_{D_{j}}f^{1/\left|D_{j}\right|}dx\leq C$$
Suppose that $\left\{f>1\right\}$ has positive measure. Then by the continuity of measure, there is $\alpha>1$ such that $\left|\left\{f>\alpha\right\}\right|=:A>0$. By the intermediate value theorem of non-atomic measures (Sierpinski's theorem), for each $j\in\mathbb{N}$, a measurable set $D_{j}\subset \left\{f>\alpha\right\}$ with
$$\left|D_{j}\right|=\dfrac{A}{2j}$$
Using the hypothesis again, we see that
$$C\geq \int_{D_{j}}f^{1/\left|D_{j}\right|}dx\geq\alpha^{1/\left|D_{j}\right|}\left|D_{j}\right|=\alpha^{2j/A}\dfrac{A}{2j},$$
where the RHS tends to $\infty$ as $j\rightarrow\infty$ since $\alpha>1$. We conclude that $A=0$. I would be curious to know if there's a way to show that show this last result without appealing to Sierpinski's theorem.