I am having some trouble with Lebesgue's Dominant Convergence theorem. It seems as if I have a counterexample, and I can't find my mistake.
Say that $\mu$ is a uniform measure over $\mathbb{N}\cup{0}$, which means $\mu(A) = |A| $.
Now, for every $x \in \mathbb{R}$, define the functions $f_x:\mathbb{N} \rightarrow \mathbb{R}$ as follows:
$$f_x(n) = \frac{x^n}{n!}$$
Obviously, $$\int_{\mathbb{N}\cup0}f_xd\mu = \sum_{n\ge0}\frac{x^n}{n!} = e^x$$
Notice that for every $n$, $\lim\limits_{x\to 0}f_x(n) = 0$, which means $\lim\limits_{x\to 0}f_x=0$ everywhere.
Also for $-1<x<1$, $|f_x|\le f_1$ and $f_1$ is summable : $\int_{\mathbb{N}\cup0}f_1 d\mu = e <\infty$
Now using LDCT, I conclude that
$$\lim\limits_{x\to 0} e^x = \lim\limits_{x\to 0}\int_{\mathbb{N}\cup0}f_xd\mu=\int_{\mathbb{N}\cup0}\lim\limits_{x\to 0}f_xd\mu = \int_{\mathbb{N}\cup0}0 d\mu = 0$$
which of course is false.
Where am I mistaken?
It's NOT true that for every $n$, $\lim_{x\to0} f_x(n)=0$. In particular, it's not true for $n=0$.