Integrating with respect to an MCMC Kernel

I'm currently trying to evaluate an integral with respect to an MCMC kernel, and I wanted someone else to read what I `proved' to see if it's really correct. It just doesn't feel right.

Suppose I have an MCMC Kernel K(a'|a) with stationary distribution p(a). I want to take the expectation of a function f(a) given a point a' through the kernel like so,

int_{a} f(a) K(a|a')

I use detailed balance,

= int_{a} f(a) frac{K(a'|a) p(a)}{p(a')} = mathbb{E}[f(a)K(a'|a)] frac{1}{p(a')}

Independence (at least I think they should be independent?)

= mathbb{E}[f(a)] mathbb{E}[K(a'|a)] frac{1}{p(a')}

And detailed balance once again

= mathbb{E}[f(a)] p(a') frac{1}{p(a')}

Now this just seems incorrect. If I interpret this correctly, it's telling me that if I take a large number of samples from $K(a|a')$ from any starting point $a'$ and integrate a function wrt to it, I might as well have had the original distribution. I must have done something wrong.