Is $g f_n$ convergentin measure if $f_n$ is convergent in measure to $0$?

Question

Let a seguence $f_n$ of real valued measurable functions on a space $X$ with a $\sigma$-finite measure $\mu$ be convergent in measure to $0$. Let a function $g$ be measurable. Is then $gf_n$ convergent to $0$ with respect to measure $\mu$ ?

Answer 1

True for finite measures but false with just $\sigma-$ finiteness. [For finite measures use my comment above].

On the real line with Lebesgue measure let $f_n(x)=\frac 1n $ for $n <x<n+1 $ and $0$ elsewhere. Let $g(x)=x$ for all $x$. Then $f_n \to 0$ in measure but $f_ng>1$ on $(n,n+1)$.