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- I'm confused over a claim made in my lecture notes, namely that the collection of finite unions of half-open intervals in ##\mathbb R## forms an algebra.
I'm reading in these notes the following passage (I only have a question about the last two sentences):
The last two sentences confuse me. Which sets does the author have in mind? I know what an algebra is (basically a sigma algebra, but not closed under countable infinite operations, but finite operations), but I'm unsure which sets the author has in mind. Specifically, is it true that the collection of finite unions of half-open intervals forms an algebra?
Looking at Folland's book, he claims that the collection of finite disjoint unions of half open intervals of the form ##\emptyset## or ##(a,b]## or ##(a,\infty)##, where ##-\infty\leq a <b<\infty##, forms an algebra, but my notes claim that the union doesn't have to be disjoint to form an algebra. Grateful for any clarification.
We briefly consider a generalization of one-dimensional Lebesgue measure, called Lebesgue-Stieltjes measures on ##\mathbb{R}##. These measures are obtained from an increasing, right-continuous function ##F: \mathbb{R} \rightarrow \mathbb{R}##, and assign to a half-open interval ##(a, b]## the measure $$\mu_{F}((a, b])=F(b)-F(a) .$$ The use of half-open intervals is significant here because a Lebesgue-Stieltjes measure may assign nonzero measure to a single point. Thus, unlike Lebesgue measure, we need not have ##\mu_{F}([a, b])=\mu_{F}((a, b])##. Half-open intervals are also convenient because the complement of a half-open interval is a finite union of (possibly infinite) half-open intervals of the same type. Thus, the collection of finite unions of half-open intervals forms an algebra.
The last two sentences confuse me. Which sets does the author have in mind? I know what an algebra is (basically a sigma algebra, but not closed under countable infinite operations, but finite operations), but I'm unsure which sets the author has in mind. Specifically, is it true that the collection of finite unions of half-open intervals forms an algebra?
Looking at Folland's book, he claims that the collection of finite disjoint unions of half open intervals of the form ##\emptyset## or ##(a,b]## or ##(a,\infty)##, where ##-\infty\leq a <b<\infty##, forms an algebra, but my notes claim that the union doesn't have to be disjoint to form an algebra. Grateful for any clarification.
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