algebraic geometry

## The Functor of Points Revisited

Mike Hopkins is giving the Milliman Lectures this week at the University of Washington and the first talk involved this idea that I'm extremely familiar with, but am also surprised at how unfamiliar most mathematicians are with it. I've made almost this exact post several other times, but it bears repeating. As I basked in… Continue reading The Functor of Points Revisited

analysis

## Applying Covering Theorems

I've searched far and wide to not do one of the standard applications that are in all grad analysis texts (yes I'm referring to the Hardy-Littlewood maximal function being weakly bounded). We are getting into the parts of analysis that I despise (it will all be over in 4 days...I hope *crosses fingers*). Claim: If… Continue reading Applying Covering Theorems

analysis

## Covering Theorem (we use past Lemmas)

A brief break occured while I moved 2700 miles away. The important thing is I'm back, and we're going to prove a big one today. First let's define a Vitali covering. A set is Vitali covered by the collection of sets $latex \mathcal{V}$ if for any $latex \epsilon>0$ and any x in the set, there… Continue reading Covering Theorem (we use past Lemmas)

analysis

## Covering Lemma 2

Today I'll do probably the best known Vitali Covering Lemma. I'll take the approach of Rudin. Statement (finite version): If W is the union of a finite collection of balls $latex B(x_i, r_i)$ (say to N), then there is a subcollection $latex S\subset \{1,\ldots , N\}$ so that a) the balls $latex B(x_i, r_i)$ with… Continue reading Covering Lemma 2

analysis

## Measure Decomposition Theorems

Well, I've been mostly posting comments around on other people's blogs and not really getting around to my own. I'm giving up on NCG for now. It seems that the stuff I already know I'm reading, and I'm skipping the stuff that will take effort to sort through. This seems pointless, especially with Analysis prelims… Continue reading Measure Decomposition Theorems

analysis

## Lebesgue Points

Just a quick detour. I've found a new reason to dislike analysis. I'm trying to learn Radon-Nikodym derivatives (i.e. an attempt to take derivatives in a general measure theory sense and maintain the Fundamental Theorem of Calculus for the Lebesgue integral), and Rudin uses the approach of Lebesgue Points. Since I've never learned this before,… Continue reading Lebesgue Points

analysis

## Lp Space Property

Something has been bothering me for a few days. Here goes. An $latex L^p$ space is a space of functions consisting of $latex \{f: \int_X |f|^pd\mu <\infty\}$. So essentially, if we can integrate the p-th power of the function and get a finite answer, then the function is in the space. (The careful reader will… Continue reading Lp Space Property