Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing notion $\mathbb{P}\_G$: for $A, B \in \mathcal{B}$ let $A \sim B \iff m(A \bigtriangleup B) = 0$, and let $\mathbb{P}\_G$ consists of equivalence classes $[A]\_\sim$ where $A \in \mathcal{B}$ and $m(A)>0$, with $[A]\_\sim \leq [B]\_\sim$ iff $m(A\setminus B)=0.$ 


Now define an equivalence relation $\equiv$ among infinite compact Abelian groups by


$G \equiv H \iff \mathbb{P}\_G$ is forcing equivalent to $\mathbb{P}\_H$.


Now my question is the following:



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> **Question 1.** What non-trivial facts one can say about $\equiv$ relation$?$ 
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As a sample of explicit question, one may ask the following:



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> **Question 2.** Let $\kappa$ be an infinite cardinal and let $\mathfrak{g}\_\kappa$ consists of compact abelian groups of size $\kappa.$ What is the cardinality of $\{[G]\_\equiv : G \in \mathfrak{g}\_\kappa \}?$
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In general one can ask similar question for probability spaces. Let $(\Omega, \mathcal{F}, P)$ be an infinite [probability space](https://en.wikipedia.org/wiki/Probability_space) and assign to it, its natural forcing notion $\mathbb{P}\_{(\Omega, \mathcal{F}, P)}$. Also define $\equiv\_P$ relation between infinite probability spaces as above.



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> **Question 3.** What non-trivial facts one can say about $\equiv\_P$ relation$?$
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Similar to question 2, one may ask:



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> **Question 4.** Let $\kappa$ be an infinite cardinal and let $\mathfrak{p}\_\kappa$ consists of probability spaces of size $\kappa.$ What is the cardinality of $\{[G]\_\equiv : G \in \mathfrak{p}\_\kappa \}?$
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**Remark.** The equivalence of forcing notions is defined in the comments by Professor Hamkins.



Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give
If $G$ is a compact group with infinite weight, then the Maharam type of the Haar measure on $G$ is equal to its weight - See Theorem 2.4 in S. Grekas, On products of topological measure spaces, Handbook of measure theory, Vol. 1, edited by E. Pap, Elsevier 2002.


As the measure algebra of a compact group is Maharam homogeneous, Maharam's theorem implies that it is isomorphic to the usual measure algebra on $2^\kappa$, where $\kappa$ is the Maraham type. This is the usual complete Boolean algebra for adding $\kappa$ random reals.


Therefore: $G \equiv H$
 (in your notation) iff $w(G)=w(H)$.


