Cylinder events

If E is countable. Let F be the power set of E. For some event A in F, is it true that the B_n below approximates A (as a Caratheodory extension)?

B_n = { [w(0),w(1),w(2),..,w(n)] : w in A }

I mean stronger, do B_n in a set theory sense converge to A?

flag	asked Oct 10 at 18:21 Mathstack 1●1

	No. e.g. $A=\lbrace x\in\lbrace 0,1\rbrace ^{\mathbb N}\colon $x has density 1/3 of 1's$\rbrace$. – Anthony Quas Oct 10 at 18:29
	Can you be a bit more precise? The Caratheodory extension is about measures, but I see no measure here. And what is convergence in a set theory sense? – Michael Greinecker Oct 10 at 22:10