Given the following Theorem from our script in measure theory:
Can anyone provide me an example of a cauchy sequence of integrable functions $f_n \in \mathcal L^1(\mu)$ where $\mu$ is a measure and a function $f \in \mathcal L^1(\mu)$ such that the equation 1.41 holds but the sequence $f_n$ does not converge almost everywhere to $f$. (We know there is a subsequence, which will converge to $f$ a.e.)
I expect there is a pathologic example of $(X, \mathcal A, \mu)$ where to find such an example, but I can't find any. Thanks in advance!
Best Answer
There is a standard example of a sequence which converges in measure but not almost everywhere. This example serves your purpose too. Consider $[0,1)$ with Lebesgue measure. Arrange the intervals $[\frac {i-1} {2^{n}}, \frac i {2^{n}})$ in a sequence by first listing the ones with $n=1$, and then the ones with $n=2$ and so on. Let $f_n$ be the characteristic function of the $n-$ th interval in this arrangement. This sequence converges in $L^{1}$ but there are infinitely many $0$'s and infinitely many $1$'s in $(f_n(x))$ for any $x$. so $(f_n)$ does not converge at any point.
The sequence converges in $L^{1}$ to $0$ (and hence it is also Cauchy) because it converges in measure and it is uniformly bounded.