Let $\Omega$ be a sample space and $\mathbb{Q}=(\Omega, \mathcal{F}, \mathbb{P})$ standard probability space. Suppose that there exeist three different functions, say
$$\mu:\Omega\to\Delta(X)$$
$$\tau:X\to\Delta(T)$$ and $$\phi:X\times Y\to \Delta(\Xi)$$
they induce a probability distribution on $X\times Y\times \Xi$ and $\mu_{\Xi}(\omega)$ is the marginal distribution on $\Xi$.
What is the formula of the marginal distributuion $\mu_{\Xi}(\omega)$ and the other marginals with respect to $X$ and $Y$ and what is the formula of the joint pdf?