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The rest of the chapter on -algebras goes through the construction of various other measurable spaces such as those on the space of 1) continuous functions, 2) functions continuous on the right, and 3) direct products of measurable spaces. The … Continue reading
Show that the following is not a Borel set in (this is the -algebra of functions over the domain unlike the previous onces we looked at where the domain was over the natural numbers). If is a Borel set, then … Continue reading
Following on from the previous post, is the following a Borel set? This is a Borel set because we can intersect the set of converging sequences and the set of sequences bounded from below.
Continuing the last post on showing that certain sets are Borel sets, today we ask if the following is a Borel set. This is the set of all sequences converging to a finite limit. My initial thought was to use … Continue reading
The chapter on measurable spaces introduces a -algebra over the real numbers . The Borel algebra, , is the smallest -algebra where is the algebra generated by finite disjoint sums of intervals of the form . By the direct product … Continue reading
Let be a countable decomposition of and be the -algebra generated by . Are there only countably many sets in ? The answer is no. We can show this by showing that the natural numbers is a strict subset of … Continue reading
The past few problems looked at how probability works when we have an infinite sample space. It didn’t cover how one can actually assign probabilities to such spaces. That will be the next task. Before that, the book covers the … Continue reading
Let be a finite measure on algebra , for and (i.e. ). Show that . Since ,
Let be a finitely additive measure on an algebra , and let be pairwise disjoint and satisfy . Then show that .