On the measure theoretic structure of compact groups

If $G$ is a compact group with $w(G)=a\geq \omega$, we show the following results:

(i)

There exist direct products ${\prod _{\xi<a}G_{\xi}, \prod _{\xi<a}H_{\xi}}$ of compact metric groups and continuous open surjections ${\prod _{\xi<a}G_{\xi} \stackrel{p}{\rightarrow }G \stackrel{q}{\rightarrow }\prod _{\xi<a}H_{\xi}}$ with respect to Haar measure; and

(ii)

the Haar measure on $G$ is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups.

Applications of the above results in measure theory are given.