# Fatou's Lemma

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$ answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss Fatou's Lemma and solve a problem from Rudin's Real and Complex Analysis (a.k.a. "Big Rudin").

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# The Borel-Cantelli Lemma

Today we're chatting about the Borel-Cantelli Lemma. When I first came across this lemma, I struggled to understand what it meant "in English." What does $\mu(\cup\cap E_k)=0$ really signify?? There's a pretty simple explanation if $(X,\Sigma,\mu)$ is a probability space, but how are we to understand the result in the context of general measure spaces?

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# Open Sets Are Everything

In today's post I want to emphasize a simple - but important - idea in topology which I think is helpful for anyone new to the subject, and that is: Open sets are everything! What do I mean by that? Well, for a given set $X$, all the properties of $X$ are HIGHLY dependent on how you define an "open set."

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# On Connectedness, Intuitively

Today's post is a bit of a ramble, but my goal is to uncover the intuition behind one of the definitions of a connected topological space. Ideally, this is just a little tidbit I'd like to stash in The Back Pocket. But as you can tell already, the length of this post isn't so "little"! Oh well, here we go!

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