Some properties of the normal image of convex functions

Let $z$ be a convex function defined in a convex domain $D$ of a finite-dimensional Euclidean space. Denote by $z^{(n)}$ the convolutions of $z$ with elements of a $\delta$-type sequence of test functions and let $\nu_z$ and $\nu_{z^{(n)}}$ be the measures of normal images corresponding to $z$ and $z^{(n)}$. One of the main results of this work is that $\nu_{z^{(n)}}\to\nu_z$ in variation on a compact $K\subset D$ if and only if $\nu_z$ is absolutely continuous on $K$ with respect to Lebesgue measure.
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