Convergence in measure and in category, similarities and differences (Q264160)

Following \textit{E. Wagner} [Fundam. Math. 112, 89--102 (1981; Zbl 0386.28005)] a sequence \(f_n:[0,1]\rightarrow\mathbb{R}\) is said to converge in category to \(f:[0,1]\rightarrow\mathbb{R}\) if every subsequence of \((f_n)_{n\in\mathbb{N}}\) has a subsequence converging to \(f\) pointwise except on a set of first category. The author compares properties of this type of convergence with those of convergence in measure. In particular the following three facts are proved. (1) If \((B_n)_{n\in\mathbb{N}}\) is a sequence of subsets of \([0,1]\) having the Baire property such that \((\chi_{B_n})_{n\in\mathbb{N}}\) converges in category to \(0\) and \((r_n)_{n\in\mathbb{N}}\) is real sequence converging to \(0\), then \((\chi_{(B_n+r_n)\cap [0,1]})_{n\in\mathbb{N}}\) converges in category to \(0\). (2) There exists a sequence \((B_n)_{n\in\mathbb{N}}\) of subsets of \([0,1]\) having the Baire property such that \((\chi_{B_n})_{n\in\mathbb{N}}\) converges in category to \(0\), but \((\frac{1}{n}\sum_{n=1}^n\chi_{B_i})_{n\in\mathbb{N}}\) does not converge in category to \(0\). (3) There exists a sequence \((B_n)_{n\in\mathbb{N}}\) of subsets of \([0,1]\) having the Baire property such that \(\frac{1}{n}\sum_{n=1}^n\chi_{B_i}(x)\) converges to \(0\) except on a set of first category, but \((\frac{1}{n}\sum_{n=1}^n\chi_{B_i+3^{-i}})_{n\in\mathbb{N}}\) does not converge in category to \(0\).