6 Show that if a $\sigma$-algebra is infinite, that it contains a countably infinite collection of disjoint subsets. An immediate consequence is that the $\sigma$-algebra is uncountable.

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning t...

So by contradiction, infinite $0-1$ strings are uncountable. Can I use the fact that $\ {0,1\}$ is a subset of any sequence of countable sets $\ {E_n\}_ {n\in\mathbb {N}}$ and say the infinite product of this is …

For many infinite-dimensional vector spaces of interest we don't care about describing a basis anyway; they often come with a topology and we can therefore get a lot out of studying dense subspaces, …

De Morgan's law on infinite unions and intersections Ask Question Asked 14 years, 7 months ago Modified 4 years, 11 months ago

Are you familiar with Taylor series? Series solutions of differential equations at regular points? From what foundation/background are you approaching this problem?

Richard Thompson's groups $T$ and $V$ are well-known examples of infinite simple groups. See this answer of mine for more details, or look up the article Introductory notes on Richard Thompson's …

What do finite, infinite, countable, not countable, countably infinite mean? [duplicate] Ask Question Asked 13 years, 3 months ago Modified 13 years, 3 months ago

Just out of curiosity, I was wondering if anybody knows any methods (other than the integral test) of proving the infinite series where the nth term is given by $\frac {1} {n^2}$ converges.

4 You need to endow your infinite set with a measure such that the whole space has measure $1$ and then integrate (and hope that your function is measurable to begin with). For finite sets, the obvious …