Abstract
A family $\{\mu_\alpha\}$ of measures on a $\sigma$-field $\mathscr{B}$ on a space $X$ is "uniformly orthogonal" means that for each $\alpha, \exists H_\alpha \in \mathscr{B}$ such that $\mu_\alpha (X - H_\alpha) = \mu_\beta(H_\alpha) = 0$ if $\beta \neq \alpha$. Assuming $CH$, an example is given of an orthogonal family of measures on the Borel sets of $I^2$ such that no uncountable subfamily is uniformly orthogonal. Assuming $\sim CH + MA$, such an uncountable family obviously exists.
Citation
Dorothy Maharam. "Orthogonal Measures: An Example." Ann. Probab. 10 (3) 879 - 880, August, 1982. https://doi.org/10.1214/aop/1176993803
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