Measure Theory

From MGSA

Can you think of any interesting functors in Measure Theory? (Rieffel)
Make $L p ( - )$ into a functor. (Rieffel)
Make "measurable sets" into a functor. (Rieffel)
Quote Caratheodory's Theorem. (Rieffel)
Consider the unit interval. What can you say about the integral of a non-negative function on the unit square? (Arveson)
What is the precise statement of the Fubini theorem? (Arveson)
Consider the following definition for f a measurable function:
1. $f - 1 (A)$ is measurable for all intervals $A$ ; and
2. $f - 1 (A)$ is measurable for all Borel sets $A$ .
Are they the same? Why? (Arveson)
If $A$ is a subset of the real plane such that the intersection of $A$ with any horizontal line is countable, what can you say about the Lebesgue measure of $A$ ? (Arveson)
What if the intersection of $A$ with every line of slope one is countable? (Arveson)
Why is the map $(x,y)\mapsto (x,y-x)$ measure preserving? Is the map $(x,y)\mapsto (x,y-3x)$ ? $(x,y)\mapsto (x,3y-2x)$ ? (Arveson)
How do you know if a map is measure preserving? (Arveson)
What does Alaoglu's theorem say? What is the basic idea of the proof of Alaoglu's theorem? (Arveson)
Name somre results which use the Baire Category Theorem. (Arveson)
What is the definition of the weak- $*$ topology? (Arveson)

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