properties of distributions If $$\int_{-\infty}^\infty f dx = 1$$, with $f > 0 \forall x$, then prove or disprove: $$\int_{-\infty}^\infty \frac{1}{1 + f} dx $$ diverges.
The hint I got is to consider the measure of the set$(x:f > 1)$. May be the measure is zero thereby ensuring the divergence of integral?

Suppose that $f$ is nonnegative (you say it's a distribution?). Notice that $1=\frac{1}{1+f}+\frac{f}{1+f}$. Also $0\leq \frac{f}{1+f}\leq f$ which means $\frac{f}{1+f}$ is integrable. Since the l.h.s. is not integrable, it follows that $\frac{1}{1+f}$ is not integrable. 
