Baire measures on \([0,\Omega)\) and \([0,\Omega]\) (Q1825301)

Let X be the set of all ordinals less than the first uncountable ordinal \(\Omega\) and let \(\bar X\) be \(X\cup \{\Omega\}\), each with the order topology. Along with some other results the author proves the following theorems. Theorem 1. Every Baire measure \(\mu\) on X is semi-regular and can be expressed in exactly one way in the form \(\mu =\nu +p\gamma\), where \(\nu\) is either O or a purely discontinuous Baire measure and \(0\leq p\leq \infty.\) Theorem 2. Every semi-regular Borel measure \(\mu\) on X can be expressed in exactly one way in the form \(\mu =\nu +p\delta\), where \(\nu\) is either O or a purely discontinuous Borel measure and \(0\leq p\leq \infty.\) Theorem 5. Every Baire measure \(\mu\) on X is regular and can be expressed in exactly one way in the form \(\mu =\nu +p{\bar \gamma}\), where \(\nu\) is a Baire measure concentrated on a countable subset of X and \(0\leq p<\infty.\)