1. Density function of random variable transformation
Given $X$ a real valued random variable with a density function $f_X$. Let $g$ be a continuous function.
Does this imply, that the random variable $g(X)$ has a density function? Or do I need more assumptions, for example that $g$ has an inverse function?
2. Density function of marginal distribution
Let $\lambda(dx,dy)$ be a Radon measure with a density function and $D$ is compact. Does this imply, that the marginal distribution $\lambda(dx)=\int_D \lambda(dx,dy)dy$ has a density function too?