I'm trying to find counterexamples of Fatou's lemma when $\inf$ is replaced to $\sup$. i.e.,
counterexamples of $\int\limsup_{n\to\infty}f_kd\lambda = \limsup_{n\to\infty} \int f_kd\lambda$
I found the case of $\geq$ fails, but I don't see what I have to construct to make $\leq$ fail. I tried to find simple function $s_k$'s such that $\limsup A_k=\mathbb{R}$ and $\lambda(A_k)=1$ where $A_k$'s are measurable, but I couldn't find any. Making $\lambda(A_k)=1$ is easy but I can't handle $\limsup$ part.
I'll make it so that $\limsup A_k = [0,+\infty)$, that's slightly easier and suffices for the task.
We start with \begin{align} A_0 &= [0, 1), \\ A_1 &= [0, 1/2) \cup [1, 3/2), \\ A_2 &= [1/2, 1) \cup [3/2, 2), \\ A_3 &= [0, 1/3) \cup [1, 4/3) \cup [2, 7/3), \\ A_4 &= [1/3, 2/3) \cup [4/3, 5/3) \cup [7/3, 8/3), \\ A_5 &= [2/3, 1) \cup [5/3, 2) \cup [8/3, 3). \end{align}
I think one can guess the pattern now, we have
$$A_{\frac{m(m-1)}{2} + k} = \bigcup_{n = 0}^{m} \biggl[n + \frac{k}{m}, n + \frac{k+1}{m}\biggr)$$
for $m \geqslant 1$ and $0 \leqslant k < m$.
However, to get a sequence of simple functions with
$$\int \limsup f_k \,d\lambda > \limsup \int f_k \,d\lambda$$
it would suffice to alternate between the characteristic functions of $[0,1/2) \cup [1,3/2)$ and $[1/2,1) \cup [3/2,2)$.
But, Fatou's lemma says that for nonnegative measurable $f_k$ we have
$$\int \liminf f_k \,d\lambda \leqslant \liminf \int f_k \,d\lambda\,,$$
not that we have equality. Thus the corresponding $\limsup$ version would state that
$$\limsup \int f_k \,d\lambda \leqslant \int \limsup f_k \,d\lambda\,.\tag{$\ast$}$$
Thus the above examples are not counterexamples to the $\limsup$ version $(\ast)$ - which does actually hold and is useful given the additional hypothesis that the sequence $f_k$ has an integrable dominating function. The failure of $(\ast)$ without that additional hypothesis is exemplified by $f_k = \chi_{[k,k+1)}$ when we have
$$\int \limsup f_k \,d\lambda = \int 0 \,d\lambda = 0 < 1 = \limsup \int f_k \,d\lambda\,.$$