Proving a product of sequences does not converge in measure to 0

Question

I have a candidate counter-example to show that on a set of infinite measure, a product $\{f_{n} g_{n} \}$ of sequences $\{f_{n} \} \to f$ in measure and $\{g_{n} \} \to g$ in measure doesn't necessarily converge in measure to $fg$. Other counter-examples I've found are very un-intuitive, so if I can finish confirming this one works it will be helpful: Consider sequence $\{f_{n} \} = \frac{1}{n} \chi_{[0, n]}$, which clearly converges in measure to function $f(x) = 0$, and sequence $\{g_{n} \} = x$ for each $n$, which converges in measure to function $g(x) = x$. It feels intuitively like the product $\frac{x}{n} \chi_{[0, n]}$ does not converge in measure to $fg = 0$, but how do I prove it rigorously?

Answer 1

For each $n$, $\frac{x}{n} \chi_{[0, n]} > 0.5$ for all $x$ in the interval $(0.5n, n)$, a set of measure $0.5n$. Hence, checking the definition of convergence in measure, your counter-example does seem to work to prove your statement.