Let
${{({{\mathcal{M}}_{i}})}_{i}}{{_{\in }}_{I}},{{({{\mathcal{N}}_{j}})}_{j\in J}}$
be families of von Neumann algebras and
$\mathcal{U},\,{\mathcal{U}}'$
be ultrafilters in
$I$
,
$J$
, respectively. Let
$1\,\le \,p\,<\,\infty $
and
$n\,\in \,\mathbb{N}$
. Let
${{x}_{1}},...,{{x}_{n}}\,\text{in}\,\prod {{L}_{p}}({{\mathcal{M}}_{i}})$
and
${{y}_{1}},...,{{y}_{n\,}}\text{in }\Pi \,{{L}_{p}}({{\mathcal{N}}_{j}})$
be bounded families. We show the following equality
$$\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}=\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}$$
For
$p\,=\,1$
this Fubini type result is related to the local reflexivity of duals of
${{C}^{*}}$
-algebras. This fails for
$P=\infty $
.