Meaning of stationary invariant measure in term of the pdf of the measure.

Question

I am starting to learn about Markov chains for continuous state spaces. In particular if we denote the space X and the transition kernel $p : X\times B(X)\to \mathbb{R}$ where $B(X)$ are the borelians on $X$ (or something like that), then we say that a measure $\pi$ on $B(X)$ is invariant if $$\pi(A)=\int_XP(x,A)\pi(dx).$$ Now suppose $\pi$ has a pdf $f$, i.e. $\pi(A) = \int_A f(x)dx$, then I think that we have $$\int_A f(z)dz = \int_X P(x,A)f(x)dx.$$ But is there a condition directly on the density $f$ for $\pi$ to be invariant (without the integral)? I.e. $\pi$ is invariant iff $$f(z) = \ldots$$?