Let $$\langle f(x,y)\rangle = \iint_S f(x,y)\,\mathrm{d}x\,\mathrm{d}y$$
I have seen the above in multiple papers as the definition of $\langle f(x,y)\rangle$. I would normally associate angle brackets with being an inner product $\langle f,g \rangle$ of two functions $f$ and $g$, but the definition I have quoted appears to be acting on a single function. Is this still defining $\langle f(x,y) \rangle$ as an inner product, or is it borrowing the notation of angle brackets resulting in my confusion?