# Integration

There have been 2 completed talks and 2 topic suggestions tagged with integration.

## Completed Talks

### Integrability of Riccati Equations

Delivered by Letian Chen on Friday January 27, 2017

In 1841, 3 years before Joseph Liouville discovered transcendental numbers, Joseph Liouville showed that the Riccati equation $y' = ay^2 + bx^m$ has a quadrature solution if and only if $m = 0, -2, -\frac{4n}{2n±1}$. Nowadays, we see Liouville's results as the foundation to qualitative analysis to differential equations, a fascinating subarea of DEs, which studies for example the existence and uniqueness of different kinds of equations. In my talk, I will (hopefully) prove the classical result of Liouville after a quick review of history and related linear theory.

### Integration

Delivered by Gregory Patchell on Friday October 28, 2016

The first part of this talk will be concerned with a brief introduction to measure theory. We will answer questions such a: How do we measure sets? Can every set be measured?

The second part of this talk will be the construction of the Lebesgue integral along with basic properties of the Lebesgue integral, along with comparisons to Riemann integration as we go. From Math 148, we saw that pointwise convergence doesn’t play well with Riemann integration. We will see that with the powerful Monotone Convergence Theorem and the Dominated Convergence Theorem, pointwise convergence and Lebesgue integration are a match made in heaven.

### Outline

• Definitions and examples of σ-algebras, measures, and measurable functions

• Motivations for Lebesgue integrals

• Construction of the Lebesgue integral with comparison to construction of the Riemann integral

• Benefits and properties of Lebesgue integration and limitations of Riemann integration

• Limitations of Lebesgue Integration

• Probability Spaces

• Vitali Sets

## Talk Suggestions

### Integrals and Graph Theory

Possible reference materials for this topic include