What do the square bracket signify in $\int [\text{d}\pi]f(\pi)$

Question

I am reading this paper which repeatedly includes integrals such as,

$$ P_M(\phi \to \phi') = \int [\text{d}\pi][\text{d}\pi'] P_G(\pi)\delta((\phi, \pi) - (\phi'', \pi'')) $$

Note that $P_G$ is a probability and $\pi$ is a momentum not $3.14\dots$ also note that $\delta(x)$ is the dirac delta function.

The integral is defined over phase space on a lattice. $\phi$ is some field.

Answer 1

It's very likely that they mean the floor function $\lfloor x \rfloor$, which yields the largest integer less than $x$. This seems even more likely as this has to do with estimates over lattices, and so should be discretized.

Then these integrals should be interpreted as Riemann-Stieltjes Integrals.