I'll be reading through Walter Rudin's Real and Complex Analysis (slowly) and keeping some short notes. This also serves as an excuse for me to test out KaTeX :) I'll either continually extend this page, or make a new post for each chapter, we shall see. Regardless, they will probably be brief and meant more of as a reminder of what it is in each chapter, but perhaps they may still be useful to someone!

Chapter 1

The chapter begins with the following:

  • Basic definitions of topological and metric spaces, \(\sigma\)-algebras and measurable functions.
  • Definition of Borel sets as the smallest \(\sigma\)-algebra containing all the open sets (therefore containing all \(F_\sigma\) and \(G_\delta\) sets).
  • Equipping a \(\sigma\)-algebra with a measure and some basic properties.
  • Defining the Lebesgue integral for measurable simple functions, and then for arbitrary measurable functions by approximating them by simple functions (from below), which is always possible.

After this, it goes into various basic properties of the Lebesgue integral (and how convenient it is!). Some important theorems are covered:

Lebesgue's Monotone Convergence Theorem: Let \(f_n\) be a sequence of measurable functions on X, and suppose that

  • \(0 \leq f_1(x) \leq f_2(x) \leq \cdots \leq \infty \)
  • \(f_n(x)\rightarrow f(x)\) as \(n\rightarrow \infty\)

Then \(f\) is measurable and

$$\int_X f_n\ d\mu \rightarrow \int_X f\ d\mu$$ as \(n\rightarrow\infty\).

Fatou's Lemma: If \(f_n: X\rightarrow [0,\infty]\) is measurable, then

$$\int_X \left(\liminf_{n\rightarrow\infty} f_n\right)\ d\mu\leq\liminf_{n\rightarrow\infty}\int_X f_n\ d\mu.$$

Change of Measure: Suppose \(f: X\rightarrow [0,\infty]\) is measurable, and

$$\varphi(E)=\int_E f\ d_\mu\qquad(E\in \mathcal{M}),$$

Then \(\varphi\) is a measure on \(\mathcal{M}\), and

$$\int_X g\ d\varphi = \int_X gf\ d\mu$$

for every measurable \(g\) on \(X\) with range in \([0,\infty]\).

Lebesgue's Dominated Convergence Theorem: Suppose \(f_n\) is a sequence of complex measurable functions on X such that

$$f(x) = \lim_{n\rightarrow\infty}(x)$$

exists for every \(x\). If there is a function \(g\in L^1(\mu)\) such that \(|f_n(x)|\leq g(x)\), then \(f\in L^1(\mu)\),

$$\lim_{n\rightarrow\infty}\int_X |f_n-f|\ d\mu = 0,$$

and

$$\lim_{n\rightarrow\infty}\int_X f_n\ d\mu = \int_X f\ d\mu.$$

Most of these proofs are more or less appeals to the definitions.

Finally, the chapter touches on completion of a measure by defining it for sets that are 'almost equal' to sets in the \(\sigma\)-algebra. This allows us to extend the concept of a measurable function to one that is defined almost everywhere (a.e.), as a set of measure zero will not affect the integral of the function.