Intuitively understanding Fatou's lemma I learnt Fatou's lemma a while ago. I am able to prove it and use it. I know examples showing that the inequality may be strict. But I don't really have an intuitive way to understand it. Any good thoughts?

Fatou's lemma tells you that in the limit "mass" can only be lost but not generated. Let's recall the satement. If $f_n,f\geq 0$ are measurable and $f_n\to f$ pointwise a.e., then we have $\int f \leq \liminf_{n\to\infty} \int f_n$.
A classical example is $f_n= n \chi_{[0,1/n]}$ where $\int f_n=1$ for all $n$, but in the limit the mass escapes to "vertical" infinity, so it is lost, and we have that $f_n\to 0=:f$ a.e., with $\int f=0$.
The other example where, mass escapes to "horizontal" infinity, is $f_n= \chi_{[n,n+1]}$. Again $f_n$ has mass $1$, but the limit has mass $0$.
If we shut down these escape possibilities, then mass is preserved, i.e. $\int f=\lim_{n\to\infty} \int f_n$. For example, one way to do that is to assume that $f_n$ are bounded and all of them are supported on a large interval $[-M,M]$. This follows from the Dominated Convergence Theorem which gives a fairly general criterion for convergence of the integral: if $|f_n|\leq g$ where $\int g<\infty$, then the mass is preserved under the limit.
