# Measure Theory

This course starts from the question "How do you measure sets of real numbers?" This leads into a conversation and analysis about the Lebesgue outer measure, and the Lebesgue measure. Once we understand these concepts, we are then in a position to ask a next question, "How can we approximate measurable sets? And how can we approximate measurable functions?" To have a satisfying answer to these questions, we look at Littlewood's three principles, state them rigorously as theorems, and investigate their answers and related concepts. This includes Egoroff's Theorem and Lusin's Theorem. From here we use these tools to develop and study the Lebesgue integral, culminating in the Dominated Convergence Theorem. From here we study differentiation and $L^{p}$ spaces, which culminates in the Riesz-Fischer Theorem. This basically tells us that $L^{p}$ spaces satisfy a completeness property, allowing us solve broad classes of problems by way of approximations. This concludes the portion of the course dealing specifically with Lebesgue measure.