Convolution with a convex function Let $f$ a convex function, and $j$ a $C^\infty$-function with compact support in $(-\infty,0]$ such that $\int j(s) ds = 1$ and set
$$ f_n(x) = n \int_{-\infty}^0 f(x+y)j(ny)dy $$
Prove that $f_n$ is continuous, convex, converges to $f$ pointwise and $f'_n$ increases to $f'_-$.
Im currently reading Continuous Martingales and Brownian Motion and they use this lemma. The fact that $f_n$ is convex is easy from definition. Im having trouble seeing the rest of the properties, since $f$ isn't necessarily integrable, so i can't use simple mollifier - convolution techniques to use dominated convergence or other teorems (i can't even see it's differentiable).
Any help could be grateful.

Hints for the  proof when $j \geq 0$: $f_n(x)=\int_{-N}^{0} f(x+\frac t n) j(t)dt$ (by a simple change of variable) where $N$ is such that the support of $j$ is contained in $[-N, 0]$. It is quite easy to conclude that $f_n$ is continuous and converges pointwise to $f$.[Remember that $f$ is continuous].
It is well known that $f(x)=f(0)-\int_x ^{0}f_{-}'(t))dt$. Use Fubini's Theorem to conclude that $f_n'(x)=-\int_{-N}^{0} f_{-}'(x+\frac t n)dt$. Since $f'_{-}$ is increasing it follows that $f_n'$ increases to $f'_{-}$.
