Borel Measurability of a subset of $R^{\infty}$

Question

Let $(a_m)_{m\geq 1}$ be a sequence of nonnegative real numbers and consider the set $$ \Gamma=\left\{y\in\mathbb{R}^{\infty}:\sum_{m=1}^{\infty}a_my_m<\infty\right\}. $$ I want to show that this set is Borel measurable, i.e., $\Gamma\in\mathcal{B}(\mathbb{R}^{\infty})$. My approach is to consider $$ \Gamma = \bigcup_{N\geq 1}\bigcap_{M\geq 1}\left\{y\in\mathbb{R}^{\infty}:\sum_{m=1}^{M}a_my_m\leq N\right\}. $$ However, I do not know how to show that the individual sets above are measurable. Please let me know if my question is not clear enough. In the bigger context of my problem, we actually assume $(a_m)$ is summable, but I don't think it is necessary here. Thanks!

Answer 1

You did not specify any topology on $\mathbb R^{\infty}$ but I will take it to be the product topology. $$ \Gamma = \bigcup_{N\geq 1}\bigcap_{M\geq 1}\left\{y\in\mathbb{R}^{\infty}:\sum_{m=1}^{M}a_my_m\leq N\right\}. $$ is wrong. You should write

$$ \Gamma = \bigcap_j\bigcup_{N\geq 1}\bigcap_{M_1,M_2\geq N}\left\{y\in\mathbb{R}^{\infty}:|\sum_{m=M_1}^{M_2}a_my_m|\leq \frac 1 j\right\}. $$ The map $ y \mapsto \sum\limits_{m=1}^{M}a_my_m$ is a continuous function on $\mathbb R^{\infty}$ so $\{y\in\mathbb{R}^{\infty}:|\sum_{m=M_1}^{M_2}a_my_m|\leq \frac 1 j\}$ is a closed set.