Homework 5

Due Date: Saturday, March 5

1. Let \((X,\mathcal{M},\mu)\) be a measure space, let \(f\colon X\to\mathbb{R}\), and suppose that \(f^{-1}(U) \in\mathcal{M}\) for every open set \(U\subseteq\mathbb{R}\). Prove that \(f^{-1}(B)\in\mathcal{M}\) for every Borel set \(B\subseteq\mathbb{R}\).
2. Let \([a,b]\subseteq\mathbb{R}\) be a closed interval, let \(E\subseteq [a,b]\), and suppose that \(E\) is Lebesgue measurable with \(m(E) > 0\). Prove that there exists a set \(S\subseteq E\) that is not Lebesgue measurable.
3. Prove that the set \(\{\log p \mid p\text{ is prime}\}\) is linearly independent over \(\mathbb{Q}\).
4. Recall that a function \(f\colon\mathbb{R}\to\mathbb{R}\) is periodic if there exists a \(T>0\) so that \(f(x+T) = f(x)\) for all \(x\in\mathbb{R}\). Prove that there exist two periodic functions \(f\colon\mathbb{R}\to\mathbb{R}\) and \(g\colon\mathbb{R}\to\mathbb{R}\) such that \[ f(x) + g(x) \,=\, x \] for all \(x\in\mathbb{R}\).