Spaces on which every pointwise convergent series of continuous functions converges pseudo-normally

A topological space $X$ is a $\Sigma\Sigma^*$-space provided that, for every sequence $\langle f_n\rangle_{n=0}^\infty$ of continuous functions from $X$ to $\mathbb{R}$, if the series $\sum_{n=0}^\infty\vert f_n\vert$ converges pointwise, then it converges pseudo-normally. We show that every regular Lindelöf $\Sigma\Sigma^*$-space has the Rothberger property. We also construct, under the continuum hypothesis, a $\Sigma\Sigma^*$-subset of $\mathbb{R}$ of cardinality continuum.