Is this discrete analogue of Fatou's lemma valid 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.

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
$$\sum_{j=1}^\infty\liminf_{k \to \infty} a_j(k)   \leqslant \liminf_{k \to \infty }\sum_{j=1}^\infty  a_{j}(k)$$
