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Jan 23, 2022 at 14:21 comment added Joseph Van Name De Groot duality is a duality between compact Haudorff partially ordered spaces and stably compact spaces. While stably compact spaces $X$ do not have enough continuous functions from $X$ to $\mathbb{R}$, they certainly have enough upper semicontinuous and lower semicontinuous functions that are all measurable and from which one can recover the measure.
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