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?
Thank you in advance!
Best Answer
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 $f$!