An interesting simple example of this stuff is ##\phi^{4}_{2}##, i.e. scalar with quartic interaction in 2D.
So you have the free path integral measure:
$$d\mu_{C}$$
The ##C## is the covariance, in standard physicists language the two point function:
$$C(x - y) = \int{\phi(x)\phi(y)d\mu_{C}}$$
Though note to be rigorous you must use smeared values as the values of a field at a point are ill-defined:
$$\int{C(x - y)f(x)g(y)d^{2}x d^{2}y} = \int{\phi(f)\phi(g)d\mu_{C}}\\
\phi(f) = \int{\phi(x)f(x) d^{2}x}$$
You denote the measure with this because once you know the covariance you know the free path integral in full detail, i.e. different free theories will only differ by their covariance.
In general we can obtain the values of observables by integrating:
$$\langle \mathcal{O}(\phi) \rangle = \int{\mathcal{O}(\phi) d\mu_{C}}$$
So then we have the quartic interaction. If we bound it in space:
$$\int_{\Lambda}{\phi^{4}(x)d^{2}x}$$
Then the path integral measure for a quartic interaction theory is:
$$e^{-\int_{\Lambda}{\phi^{4}(x)d^{2}x}}d\mu_{C}$$
So we can compute the normalization of this measure by:
$$\langle e^{-\int_{\Lambda}{\phi^{4}(x)d^{2}x}} \rangle = \int{e^{-\int_{\Lambda}{\phi^{4}(x)d^{2}x}}d\mu_{C}}$$
However because ##\int_{\Lambda}{\phi^{4}(x)d^{2}x} = +\infty## for almost all fields since they are distributions then for almost all fields:
$$e^{-\int_{\Lambda}{\phi^{4}(x)d^{2}x}} = 0$$
And so the path integral vanishes.
(More rigorously you can show it vanishes by using Holder's inequality with integration by parts)
However if we renormalize, which is simply Wick ordering in ##d = 2##, the quartic term becomes:
$$\int_{\Lambda}{:\phi^{4}:(x)d^{2}x}$$
By Jensen's inequality for a convex function ##F## we have:
$$F(\langle A \rangle) \leq \langle F(A) \rangle$$
So
$$e^{-\langle \int_{\Lambda}{:\phi^{4}:(x)d^{2}x}\rangle} \leq \langle e^{-\int_{\Lambda}{:\phi^{4}:(x)d^{2}x}} \rangle$$
The expectation term on the left is easy to evaluate:
$$\bigg \langle \int_{\Lambda}{:\phi^{4}:(x)d^{2}x} \bigg\rangle = 0$$
It's just a simple Feynman diagram problem in the free theory.
So we have:
$$1 \leq \langle e^{-\int_{\Lambda}{:\phi^{4}:(x)d^{2}x}} \rangle$$
Thus the interacting path integral no longer vanishes. However we have to show the Wick ordering doesn't cause it to diverge.
Note how this is different from the perturbative picture. Nonperturbatively unrenormalized theories vanish. The renormalization process ensures they don't vanish, but you have to check it doesn't cause them to diverge.
