I am trying to understand how expectations of a Markov chain change under a Metropolis Hastings step. Let's say we start at initial distribution $p_0$, and move to distribution $p_1$, using a Metropolis-Hastings step whose stationary/invariant distribution is $p^*$. Then, let's say that we know the expectation of some positive function $f$, under both the initial and stationary distribution. \begin{align*} C_0 &= \int p_0(x) f(x) dx\\ C^* &= \int p^*(x) f(x) dx \end{align*}
Can I use these to bound the expectation under $p_1$ (drawn using a Metropolis-Hastings acceptance and rejection step whose invariant distribution is $p^*$). $$ \int p_1(x) f(x) dx \leq ~??? $$
My intuition is that the expectation must approach that under the invariant distribution. That is, if $C^* < C_0$, then this expectation must decrease towards $C^*$ as well. However, I am unsure how to prove this.