I just googled 'integrable tensor' but I haven't found any stuff about it. I know this question is silly but the rigid definition is also important. Thanks in advance.
I saw this word here :
Definition Let $\Omega$ be an open subset of $\mathbb{R}^d$. A divergence-free positive symmetric tensor is a locally integrable tensor $x \mapsto A(x)$ over $\Omega$ with the properties that $A(x)\in \text{Sym}_d^{+}$ almost everywhere, and $\text{Div}A=0$.
My guess : Each entries of the tensor is integrable.