Homeworks
Final Essay on some counterexamples in measure theory:
Summary of Lebesgue Integral
We define a Lebesgue Integral by introducing the notion of an undergraph of a function $f:\mathbb{R}\mapsto [0, \infty]$. We define the undergraph $U(f)$ as $\{(x, y) \in \mathbb{R} \times [0, \infty]: 0 \leq y < f(x) \}$.
We say that $f$ is measurable if $U(f)$ is measurable, and if this is the case, then we can officially define the Lebesgue Integral as $\int f = m(U(f))$.
To compare and contrast Lebesgue integration with Riemann integration, we can consider a number of different aspects:
Riemann integrals are only defined for bounded functions over bounded intervals. Now, with Lebesgue integrals, we can integrate any measurable function.
Summary of Key Steps of Results in Lebesgue Measure Theory
I will include a summary of the approach followed by Tao. We begin by introducing the notion of a measurable set, and every measurable set is assigned a Lebesgue measure. Measurable sets obey complementarity as well as the Borel property, Boolean algebra property, and sigma-algebra property.
To introduce the notion of a Lebesgue measure, we begin with defining the outer measure as the infimum of the total volume of countably many open boxes to cover the set. Outer measure obeys some but not all the properties of Lebesgue measure. However, outer measure fails countable additivity.
Measurable sets satisfy the property that if the set is used to divide any other set into two pieces, the measures of those pieces satisfy additivity. For measurable sets, the Lebesgue measure is defined, as is equal to the outer measure.
Finally, measurable functions are functions which have a measurable pre-image with respect to any measurable set.
This setup allows Tao to define Lebesgue integration in Chapter 8. By introducing simple functions, functions which have finitely many values in their images, and defining the Lebesgue integral for simple functions, we can then define the Lebesgue integral for any non-negative measurable function by taking the supremum of the integral of a simple function which is dominated by the function. From there, we can integrate any absolutely integrable function, which is a function which has a finite Lebesgue integral of its absolute value. If this is the case, we define the integral to be the integral positive parts minus the integral of the absolute value of the negative parts.
Summary of Littlewood's Three Principles
Littlewood's First Principle states that given $\epsilon > 0$, and some measurable $E \subset [a, b]$ contains a compact subset covered by finitely many intervals, and the union of these intervals differs from E by a set of measure less than $\epsilon$. Another way to state this is that the set $E$ is a finite union of intervals, except for an $\epsilon$-set.
Littlewood's Second Principle states that if you have a measurale function, then for any $\epsilon > 0$, removing a set of measure $\epsilon$ will result in a continuous function.
Littlewood's Third Principle states that if you have a sequence of measurable functions mapping an interval $[a, b]$ to $\mathbb{R}$, which converges almost everywhere, then except for a set of measure $\epsilon$, the sequence converges uniformly.
Littlewood's Second Principle is called Lusin's Theorem and Littlewood's Third Principle is called Egoroff's Theorem.