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)

$\displaystyle R_{x}(t)|_{t=0}= power = E[X^2(t)] = \sigma^2 \cdot \delta(t)|_{t=0} = infinite = Var[X(t)] = \sigma^2 = finite$

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.