# random real forcing

Let $\kappa$ be an infinite cardinal and let $B$ be the random forcing for adding $\kappa-$many random reals.

what are the elements of $B$.More precisely given a condition $p \in B$, what are the domain and the range of $p$, if there are any?. What does it mean "a coordinate of $p$"?.

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This question is more or less the same as this one: mathoverflow.net/questions/55832/what-is-random-real-forcing –  Eran Oct 28 '11 at 17:12
I think that now there is somewhat of a stronger set theorists base in math.stackexchange.com that this question can be asked there instead. –  Asaf Karagila Oct 28 '11 at 18:34
There are different ways of representing $B$. One possibility is to let $B$ consist of all closed subsets of $2^\kappa$ of positive measure; another possibility is to allow all positive Borel sets. Two conditions are equivalent if their symmetric difference has measure zero.
Each basic clopen set has a finite set of coordinates. By ccc, each open set is almost the same as some open set that uses only countably many coordinates. So every closed/Borel set $p$ is equivalent to a set that uses only countably many coordinates. I guess that these are the "coordinates of $p$".
It might be worth adding that the complete Boolean algebra associated to this notion of forcing is the algebra of Borel subsets of $2^\kappa$ modulo sets of measure 0. In particular, thanks to the ccc, this algebra really is complete, not just countably complete as one might at first think. –  Andreas Blass Oct 28 '11 at 18:56