I was solving an exercise problem in Rudin's Real and Complex Analysis when I came across this statement.
Construct an example in which Vitali's theorem applies although the hypotheses of Lebesgue's theorem [DCT] do not hold.
I am really wondering what such an example would be. UI requires $\mathcal{L}^1$ and boundedness, while DCT requires boundedness by another function. I am unable to segregate the 2 different conditions. Could someone point me to an example that behaves as stated by Rudin?
Vitali convergence theorem: $(X,\mathcal{F},\mu)$ is a positive measure space. If a) \mu(X)<\infty b) $\{f_n\}$ is uniformly integrable c) $f_n(x)\to f(x)$ a.e. as $n \to \infty$ and d) |f(x)|<\infty a.e. then 1) $f\in \mathcal{L}^1(\mu)$ 2) $\lim_{n\to \infty} \int_{X}|f_n-f|d\mu=0$.