Problem 1
Given any charge \(\Phi\) defined on a o-algebra \(\mathscr{S}\), prove that there is a constant \(\mathrm{M}>0\) such that \(|\Phi(E)|<\mathrm{M}\) for all \(\mathrm{E} \in \mathscr{S}\).
Problem 3
Prove that a charge \(\Phi\) vanishes identically if it is both absolutely continuous and singular with respect to a measure \(\mu\).
Problem 6
Prove that if a charge \(\Phi\) is absolutely continuous (with respect to a measure \(\mu\) ), then so are its positive, negative and total variations.
Problem 8
Let \(X\) be the square \(0
