# Isomorphism of Measure Spaces

## Contents

## Personal Motivation

The reason I become interested in this category was the following. We learned that the standard topology on is generated by a basis. This basis is the collection of all open intervals . Now, consider the collection A of subsets in : . The topology generated by the half-open intervals, called the lower limit topology, yields a different topology on the real numbers than the standard topology. We can see this by recalling that the standard topology is connected but the lower limit topology is not connected. So now consider Theorem 1.18 from[1],

If , then

This construction sort of has the same flavor as our topology example. But in this case, we get a positive result.

## Definition

Let be a measurable space and a sigma algebra on . Similary, Let be a measurable space and a sigma algebra on . Let and be measurable spaces.

- A map is called measurable if for every .

- These two measurable spaces are called isomorphic if there exists a bijection such that and are measurable (such is called an isomorphism).

## Theorem

[If is a topological space, then denotes the sigma algebra of Borel subsets of .]

Let and be Borel subsets of complete separable metric spaces. For the measurable spaces and to be isomoprhic, it is necessary and sufficient that the sets and be of the same cardinality.

## Properties

We seek to find maps that preserve "essential" structures between measure spaces. Intuitively, we want at the minimum, maps to send sets of measure zero to sets of measure zero.

### Smooth maps send sets of measure zero to sets of measure zero

Let be an open set of , and let be a smooth map. If is of measure zero, then is of measure zero.

### Mini-Sards Theorem

Let be an open set of , and let be a smooth map. Then if , has measure zero in .

## Example

Consider where . is easily seen to be a smooth map since it has partial derivatives of all order. Let . Pick . Consider the cover . Then is covered by the union of all . Now, . Since was aribtary .

## Reference

1. Folland, Gerald. B; "Real Analysis: Modern Techniques and Their Applications." Wiley. 2007.

2. Victor Guillemin, Alan Pollack; "Differential Topology." Prentice-Hall. 1974.