Let $\{A_n\}$ be a sequence of sets $A_n\subset X$ of finite Lebesgue measure $\mu$ with the property that$$\forall\varepsilon>0\quad\exists N\in\mathbb{N}^+:\forall n,m\geq N\quad\mu(A_n\triangle A_m)<\varepsilon$$ I know that, if $\mu(X)<\infty$, then there is a measurable set $A$ which is the limit of the sequence, i.e. for any $\varepsilon>0$ there exists a natural $M$ such that $\forall m\geq M\quad\mu(A_n\triangle A)<\varepsilon$.

I suspect that, if $\mu(X)=\infty$, such a set could have infinite measure $\mu(A)=\infty$, but I cannot find an example of that. Does anybody know more about it? I $\infty$-ly thank you!

  • 2
    $\begingroup$ If all the $A_n$ have finite measure and $A$ has infinite measure, you will have $\mu(A_n \triangle A) = \infty$ for every $n$, so convergence is impossible. $\endgroup$ – Nate Eldredge Nov 10 '14 at 15:17

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