Let $g:R→R$ be a non-negative integrable function. Let $f:R→R$ be a bounded measurable function satisfying $f(x)>1$ for every $x∈R$. Suppose that $∫_R f^n g≤M$ for every $n∈N$. Show that $g(x)=0$ for almost all $x∈R$.
I think can use the next theorem:
Let ${f_n}$ be a sequence of measurable functions on a set of finite measure $E$. Suppose ${f_n}$ is uniformly pointwise bounded on E,that is, there is a number $M≥0$ for which $|f_n|≤M$ for all $n$. If ${f_n}→f$ pointwise on E, then $lim_{n→∞} ∫_E f_n=∫_E f$.