This question is about a passage in Reed & Simon concerning the Wiener measure and path integrals. To give the context, the authors consider the Trotter product formula
$$e^{-t(H_0+V)}f=\lim_{m\to \infty}(e^{-(t/m)H_0}e^{-(t/m)V})^mf$$
Where
$$(e^{-(t/m)H_0}e^{-(t/m)V})^mf=\int\cdots \int p\left(x_0,x_1;\frac{t}{m}\right)e^{-(t/m)V(x_1)}\cdots p\left(x_{m-1},x_m;\frac{t}{m}\right)e^{-(t/m)V(x_m)}f(x_m) dx_1\cdots dx_m\tag{X.119}$$
Where $p(x,y;t)$ is defined by $$p(x,y;t)=(4\pi t)^{-d/2}e^{-|x-y|^2/4t}$$
The authors then say:
There are two problems in trying to take $m$ to infinity in the right hand side of (X.119). The first is that the infinite product of Lebesgue measures does not yield a reasonable measure.
Now, this might be very silly and it might be a very basic thing, but why the infinite product of Lebesgue measures does not yield a reasonable measure?
What is the intuitive picture here and what is the rigorous justification?