Let \(w: \mathbb{R} ightarrow \mathbb{R}\) be an increasing (and hence measurable, by Problem 8.21 ) and bounded function. Show that for every positive \(u \in \mathcal{L}^{1}\left(\lambda^{1}ight)\) the convolution \(u \star w\) is again increasing, bounded and continuous.

Data from problem 8.21

Show that every increasing function \(u: \mathbb{R} ightarrow \mathbb{R}\) is \(\mathscr{B}(\mathbb{R}) / \mathscr{B}(\mathbb{R})\)-measurable. Under which additional condition(s) do we have \(\sigma(u)=\mathscr{B}(\mathbb{R})\) ?

[ show that \(\{u<\lambda\}\) is an interval by distinguishing three cases: \(u\) is continuous and strictly increasing when passing the level \(\lambda\); and \(u\) jumps over the level \(\lambda\); and \(u\) is 'flat' at level \(\lambda\). Draw a picture of these situations.]