Let $R^{\alpha}$ be a vector space of all real-valued functions defined on a non-empty parameter set $\alpha$. Let $\cal{B}( R^{\alpha})$ denotes a Borel $\sigma$-algebra of subsets of $R^{\alpha}$ generated by Tychonoff topology. Let a vector subspace $R^{(\alpha)}$ be defined as follows: $$ R^{(\alpha)}=\{(x_i)_{i \in \alpha}:\mbox{card}\{i : x_i\neq 0\}<\omega \}. $$ Obviously, $R^{(\alpha)}$ is a vector space of all finite $\alpha$-sequences.
Question 1. Does there exist a non-zero $\sigma$-finite $R^{(\alpha)}$-invariant Borel measure in $R^{\alpha}$?
Remark 1. An answer to Question 1 is yes when $\mbox{card}(\alpha) \le \omega$.
Remark 2. An answer to Question 1 is no when $\mbox{card}(\alpha) > \omega$ and we require that $\mu$ is Radon.
