Applying the dominated convergence theorem back and forth

Question

Suppose $(X,\mathcal{M},\mu)$ is a measure space, $(f_{n})$ and $(g_{n})$ two sequences of integrable functions that tend to $f(x)$ and $g(x)$, respectively. Suppose also that $$|f_{n}g_{n}|\leq h_{1},$$ $$|f_{n}|\leq h_{2}$$ for $h_{1}$ and $h_{2}$ integrable so that we can use dominated convergence theorem for $ f_{n}g_{n} $ and for $ f_{n} $. Is it correct to apply back and forth dominated convergence theorem and write: \begin{align*} \lim_{n \to \infty}\int f_{n}g_{n}d\mu&=\int\lim_{n \to \infty}f_{n}g_{n}d\mu\\ &=\int\lim_{n \to \infty}f_{n}\lim_{n \to \infty}g_{n}d\mu\\ &=\int\lim_{n \to \infty}f_{n}g d\mu\\ &=\lim_{n \to \infty}\int f_{n}g d\mu? \end{align*}

Answer 1

There is no reason why $\int f_ng$ should even exist. On $(0,1)$ with Lebesgue measure let $f_n\equiv \frac 1 n, g_n(x)=\frac 1 x I_{(\frac 1 n, 1)}$. Then $f_n \to f=0$, $g_n \to g$ where $g(x)=\frac 1 x$ for all $x$ and the assumptions are satisfied with $h_1=h_2=1$. But $\int f_ng$ does not exist for any $n$. So the last step in your argument is not correct . All other steps are OK.