Example of an increasing non-nonnegative sequence violating conclusion of monotone convergence theorem in space of finite measure

Question

With Lebesgue measure in $\mathbb{R}$, $f_n(x) \equiv -\frac{1}{n}$ is a good example which doesn't coincide with MCT.

However, I couldn't find another example when the measure is finite.

Could anyone give an example which MCT can't apply in finite measure space?

Thanks!

Answer 1

Is $$ (0,1)\to \Bbb{R}, x \mapsto -1/(nx) $$ what you are looking for?