Can you take a limit inside an expectation? Not always. Double your stake on a fair coin until you win and almost every outcome ends at a profit of 1, yet every stage has mean 0: the losses are packed into a set that shrinks to nothing, and the limit cannot see it. These pages show where the mass goes, which theorems stop it escaping, and why sequences can converge in different senses: almost surely, in probability or in mean.
Monotone convergence: if nonnegative functions increase to a limit, their integrals increase to the integral of the limit. Dominated convergence: if the functions converge and all stay below one fixed integrable function, the integrals converge to the integral of the limit. Both give conditions under which the limit and the expectation can be swapped. Fatou's lemma needs only that the functions are nonnegative, and it gives just an inequality: the expectation of the limit is at most the limit of the expectations.
The spike n·1[0, 1/n), which equals n on the interval from 0 to 1/n and 0 elsewhere, has mean 1 for every n, but at any fixed outcome it is eventually 0. All its mass sits in a shrinking set. In the limit the function is 0 almost surely, so its mean is 0, which is below 1. Fatou's inequality holds with room to spare. There is no integrable function above all the spikes, so dominated convergence does not apply.
On the first page switch between the five sequences and compare the mean of the limit with the limit of the means: the doubling strategy and the spike lose mass, while a dominated sequence or an increasing one do not. On the modes page compare the typewriter, the shrinking spike and a sample mean of coin flips, and see which of the three senses of convergence each one satisfies.
No, as the spike shows: it converges to 0 almost surely, but its mean stays at 1. The typewriter sequence goes the other way: it shrinks in probability and in mean while failing to converge at any single outcome. Going from almost sure convergence to convergence in mean needs a roof (domination) or uniform integrability. Convergence in mean always implies convergence in probability, but not the reverse without extra conditions.