I was curious if this discrete analog of Fatou's lemma is valid:
\begin{align} \sum_{j=1}^\infty \liminf_{k \rightarrow\infty} a_j(k) \leq \liminf_{k \rightarrow\infty} \sum_{j=1}^\infty a_j(k) , \end{align} where $a_j(k)$ is a doubly indexed sequence of real numbers.
Does it hold in the general real case? What if $a_j(k) \geq 0 $, does it hold then? Thanks to all helpers.
On
For an elementary proof (without measure) when $a_k(j)$ is nonnegative, note that for all $n \geqslant k$ we have $\inf_{m \geqslant k} a_j(m) \leqslant a_{j}(n)$ and for every positive integer $J$,
$$\sum_{j=1}^J\inf_{m \geqslant k} a_j(m) \leqslant \sum_{j=1}^J a_{j}(n) $$
Thus,
$$\sum_{j=1}^J\inf_{m \geqslant k} a_j(m) \leqslant \inf_{n \geqslant k}\sum_{j=1}^J a_{j}(n) \leqslant \inf_{n \geqslant k}\sum_{j=1}^\infty a_{j}(n)$$
Taking the limit of both sides as $k \to \infty$ yields
$$\sum_{j=1}^J\liminf_{k \to \infty} a_j(k) \leqslant \liminf_{k \to \infty }\sum_{j=1}^\infty a_{j}(k)$$
Finally, taking the limit as $J \to \infty$, we get the result
$$\sum_{j=1}^\infty\liminf_{k \to \infty} a_j(k) \leqslant \liminf_{k \to \infty }\sum_{j=1}^\infty a_{j}(k)$$
An infinite sum can be realized as an integral w.r.t counting measure on the integers. So the inequality holds in the non-negative case. Fatou's Lemma requires non-negativity and and the inequality is false without it.
$a_j(k)=-1$ for $j=k$ and $0$ for $j \neq k$ gives a counter-example.