Integral equation: averaging

Question

Let $X, Y, Z$ be Borel spaces, for simplicity we can assume that they are $\Bbb R$. Consider an equation $$ \int_{X\times Y}f(x,y,z) \kappa(x,\mathrm dy)\mu(\mathrm dx) = \int_{X\times Y}f(x,y,z) \pi(\mathrm dy)\mu(\mathrm dx) \qquad \forall z\in Z. $$ Here $f$ is a measurable function, $\mu$ and $\pi$ are probability measures, and $\kappa$ is a stochastic kernel. Intuitively it implies that $\pi$ is a certain average of $\kappa$, based on $\mu$ and $f$, that is $$ \pi(\mathrm dy) = \int_Y \kappa(x, \mathrm dy)\nu(\mathrm dx). $$ Is that true, and if it is - how can I find $\nu$?