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 is a description of "semi-continuity" of the integral operator $\int_{\Omega}(\bullet)=E(\bullet)$.
Think of the the integral operator as a mapping from a space $F_{\Omega}$ consisting of measurable functions $f:\Omega\rightarrow\mathbb{R}$. One may wonder whether the pointwise topology in this space $F_{\Omega}$ will allow the integral operator to be continuous. i.e. 
$$lim_{n\rightarrow\infty}E(f_n)=E(lim_{n\rightarrow\infty}f_n)$$
For arbitrary $f_n\in F_{\Omega}$, the best possible answer is given by Fatou's Lemma, i.e.
$$E(liminf_{n\rightarrow\infty}f_n)\leq liminf_{n\rightarrow\infty}E(f_n)\leq limsup_{n\rightarrow\infty}E(f_n)\leq E(limsup_{n\rightarrow\infty}f_n)$$
And for those $f_n$ such that $E(liminf_{n\rightarrow\infty}f_n)= E(limsup_{n\rightarrow\infty}f_n)=E(lim_{n\rightarrow\infty}f_n)$, $E(\bullet)$ possesses the continuity. Two sufficient conditions that $f_n$ belongs to this class are given by Dominate Convergence Theorem and Monotonic Convergence Theorem.
(1)Dominate Convergence Theorem actually pushes a step further, finding another $g_n\equiv |g|$ sequence $$E(-liminf_{n\rightarrow\infty}g_n)\leq E(liminf_{n\rightarrow\infty}f_n)\leq liminf_{n\rightarrow\infty}E(f_n)\leq limsup_{n\rightarrow\infty}E(f_n)\leq E(limsup_{n\rightarrow\infty}f_n)\leq E(limsup_{n\rightarrow\infty}g_n)$$ such that $E(g)<\infty$ holds.
(2)Monotonic Convergence Theorem boldly asserts that $E(liminf_{n\rightarrow\infty}f_n)= E(limsup_{n\rightarrow\infty}f_n)=E(lim_{n\rightarrow\infty}f_n)$ because monotonic convergence function sequence cannot have different upper and lower limits at any point and this suffices to guarantee their integrals to be the same.
And this intuition is the start of investigation of the lattice structure of $F_{\Omega}$.
