My graduate text on measure theory (based on these lecture notes) is now published by the AMS as part of the Graduate Studies in Mathematics series… 23 more words

## Tags » 245A - Real Analysis

#### 245A, Notes 6: Outer measures, pre-measures, and product measures

In this course so far, we have focused primarily on one specific example of a countably additive measure, namely Lebesgue measure. This measure was constructed from a more primitive concept of… 9,250 more words

#### 245A: Problem solving strategies

This is going to be a somewhat experimental post. In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a dozen or so basic tricks and techniques that are used over and over again. 5,544 more words

#### 245A, Notes 5: Differentiation theorems

Let be a compact interval of positive length (thus ). Recall that a function is said to be *differentiable* at a point if the limit… 10,897 more words

#### Covering a non-closed interval by disjoint closed intervals

The following question came up in my 245A class today:

761 more wordsIs it possible to express a non-closed interval in the real line, such as [0,1), as a countable union of disjoint closed intervals?

#### 245A, Notes 4: Modes of convergence

If one has a sequence of real numbers , it is unambiguous what it means for that sequence to converge to a limit : it means that for every , there exists an such that for all . 4,114 more words

#### 245A, Notes 3: Integration on abstract measure spaces, and the convergence theorems

Thus far, we have only focused on measure and integration theory in the context of Euclidean spaces . Now, we will work in a more abstract and general setting, in which the Euclidean space is replaced by a more general space . 8,722 more words