Variance of a white noise

Hi,

I have a pretty simple question which I thought I do not need to make a topic about, but Google is actually not helping, which is surprising. So here it goes:

How can white noise have infinite power if its variance is finite?

As far as I am aware, the following is always valid for a stationary zero-mean random process X which is classified as white noise (i.e. flat power spectrum)

assuming that the statisics of the random process are anything with the finite variance, for example, Gaussian distribution. So, yeah, I'm looking at the AWGN.

So, what gives?

Although I am aware of the physique of the realistic white processes, I am purely interested in the theoretical POV here, so I assume that this white process indeed has an infinite power. How is that possible when at the same time its probability distribution has finite variance?

Many thanks in advance.