I'm trying to solve the following exercise:
Let $f,g:\mathbb R_+\rightarrow \mathbb R_+$ be given by $f(x) = \sum_{n=0}^\infty n\mathbb 1_{[2n,2n+2)}(x)$ and $g(x) = \sum_{n=0}^\infty\mathbb 1_{[2n,\infty)}(x)$. Is it true that: a) $\sigma(f) \subset \sigma(g)$ or b) $\sigma(g) \subset \sigma(f)$?
I tried to solve it this way:
$f^{-1}([0,a]) = [0, 2b+2)$, where $b \in \mathbb N_0, b = \max\{n\in \mathbb N_0: n \le a\}, a \in \mathbb R_+\setminus\{0\} $. Therefore $\sigma(f) = \sigma(\{[0, 2b+2): b \in \mathbb N_0\})$, since $\mathcal B(\mathbb R_+) = \sigma(\{[0,a]: a \in \mathbb R_+\setminus\{0\}\}).$
In turn, $g^{-1}([1,a]) = [0, 2c), c \in \mathbb N, c = \max\{n \in \mathbb N:n \le a\}, a \in \mathbb R_+, a \gt 1$. Therefore $\sigma(g) = \sigma(\{[0,2c): c \in \mathbb N$}).
So I obtain $\sigma(f) = \sigma(g)$. Do I miss something?