## Mass Theory – Prove that \$ mathcal B ( mathbb R) ^ infty subsetneq mathcal B ( mathbb R ^ infty) \$.

I want to prove that $$mathcal B ( mathbb R) ^ infty subsetneq mathcal B ( mathbb R ^ infty)$$, from where $$mathbb R ^ infty = prod_ {i = 1} ^ infty mathbb R$$. $$mathcal B (A)$$ is the Borel $$sigma –$$Algebra of $$A$$ and $$mathcal B ( mathbb R) ^ infty$$ is the product $$sigma –$$Algebra on $$mathbb R ^ infty$$,

I do not really know how to go about it. I know that $$mathcal B ( mathbb R) ^ infty = sigma ( mathcal C)$$ from where $$mathcal$$ is the amount of cylinder, d. H. the amount of form $$C_ {i_1, …, i_n} (B_1, …, B_n) = {x in mathbb R ^ infty mid x_ {i_1} in B_1, …, x_ {i_n } in B_n }$$ from where $$B_i$$ Borel starts $$mathbb R$$,

Now I know that $$mathcal B ( mathbb R ^ infty)$$ is the $$sigma –$$Algebra generated by open sets of $$mathbb R ^ infty$$, A basis is given by $$left { prod_ {i = 1} ^ infty U_i mid U_i text {open mathbb R and} U_i neq mathbb R text {for a finite number of i } right }.$$

For the recording it is clear whether $$left { prod_ {i = 1} ^ nB_ {j_i} mid prod_ {i = 1} U_i in mathcal B ( mathbb R ^ infty) text {where} U_ {j_i} = B_ {j_i} text {and} U_k = mathbb R text {if k notin {j_1, …, j_n } } right },$$
is a $$sigma –$$Algebra, the open set of included $$mathbb R ^ n$$and thus are cylinders in $$mathcal B ( mathbb R ^ infty)$$,

How can I prove that the inclusion is strict?

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