TITLE: Random real forcing
QUESTION [5 upvotes]: Let $\kappa$ be an infinite cardinal and let $B$ be the random forcing for adding $\kappa$-many random reals.
Question: 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 $p$"?.

REPLY [4 votes]: 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 $p$".