# Follow up on understanding path integral measures

A while ago I asked the following question: Understanding Measure in Path integrals and got to the conclusion that path integral measures are infinite products of $$d\phi(x_i)$$ for some scalar field $$\phi$$.

What does this differential mean? How would one evaluate it?

The differential in a path integral means the same thing as in an ordinary integral, such as $$\int_{-\infty}^\infty d\phi\ f(\phi).$$ A "path" integral, defined on a lattice as described in Qmechanic's answer to your earlier question, is just an ordinary multi-variable integral with one integration variable $$\phi(x)$$ for each point $$x$$ in the lattice. Think of $$x$$ as an index labeling the different variables. The lattice can be very large and very fine, as long as the total number $$N$$ of sites is finite. We can have $$N=10^{100000}$$ if we like. It's still just an ordinary multi-variable integral, and the integrand is just an ordinary function of all of those variables.
If the lattice had only one point ($$N=1$$), then the "path" integral would reduce to an ordinary single-variable integral, written like this: $$\int_{-\infty}^\infty d\phi(x)\ f\big(\phi(x)\big).$$ Again, $$\phi(x)$$ is just an unusual notation for a single variable. On a lattice with a large number $$N\gg 1$$ of points (as usual), this becomes $$\int_{-\infty}^\infty d\phi(x_1)\ \int_{-\infty}^\infty d\phi(x_2)\ \cdots \int_{-\infty}^\infty d\phi(x_N)\ f\big(\phi(x_1),\phi(x_2),...,\phi(x_N)\big).$$ The collection of variables $$\phi(x_n)$$ is written to look like a function of $$x$$, because the intuition is that for a very large and very fine lattice, we might as well be working with a continuum of variables... but to define what that means, we use a lattice. (It also relies on the integrand having a special form.) The continuum limit isn't taken until after evaluating the integrals, if it's ever actually taken at all.