# Uniquness of convergence in measure

My definition of convergence in $$\mu$$-measure for the measure space $$(\Omega,\mathcal{A},\mu)$$ is:

Let $$(f_n)_n$$ and $$f$$ be measurable functions $$\Omega \to \bar{\mathbb{R}}$$. Then $$f_n$$ converges to $$f$$ in $$\mu$$-measure if $$\forall A\in\mathcal{A}$$ such that $$\mu(A)<\infty$$ and $$\forall \epsilon>0$$ we have $$\lim_{n\to\infty} \mu(A \cap\{\vert f_n-f\vert>\epsilon\})=0.$$

My book says that, if $$(\Omega,\mathcal{A},\mu)$$ is not $$\sigma$$-finite, the limit $$f$$ is in general not uniquely determined by convergence in $$\mu$$-measure.

Can you give me an example of a not unique limit?

Let $$\mu (\emptyset)=0$$ and $$\mu (A)=\infty$$ for every non-empty set $$A$$. Then any sequence $$(f_n)$$ of measurable functions converges in measure to any measurable function $$f$$!