Suppose that $f(\theta) : [0, 2\pi] \rightarrow \mathbb{R}$ is a monotone increasing real valued function and $\{k_n\}$ is an even summability kernel.
I want to show that $f \star k_n$ converges pointwise. My intuition is that the limit of this pointwise convergence is $f$, but I'm not sure how to prove this.