By random forcing, I mean the partial order of Borel sets of the given space, modulo Lebesgue null sets, ordered by inclusion. I can not find a source proving that all these partial orders are forcing equivalent. Can anyone provide a source or mention why? Thanks.



Why is Random forcing with $\mathbb{R}$, $2^\omega$, $\omega^\omega$ all the same?
Let $X, Y$ be two of the spaces you mention. Then we can construct a Borel function $f: X\rightarrow Y$ such that for some null set $N\subset X$, $f\upharpoonright X\setminus N$ is a bijection between $X\setminus N$ and $Y$.


Now given a condition $p$ in $R\_Y$ (the random forcing associated to $Y$), consider $f^{-1}(p)$. This is a condition in $R\_X$ (keep in mind that Borel functions are measurable!), and embeds $R\_Y$ into $R\_X$ densely. 


