Math 73/103
Measure Theory and Complex Analysis

Syllabus

The following is a tentative syllabus for the course. This page will be regularly updated as we go forward.

WkLecturesTopicsRemarks
19/12
  • Review of the Riemann Integral. Subdivisions, upper and lower estimates. Sets of measure zero in $\mathbb{R}$ (ex: countable subsets). A bounded function on a line segment is Riemann integrable if and only if its set of discontinuities has measure 0. Example and non-example of Riemann-integrable functions. Shortcomings of the Riemann Integral. Characterisation of the Riemann integral by Riemann sums. -PC
9/14
  • The desire to replace intervals with more general subsets when developing integration. The question of how to measure the size of such sets in a way analagous to the length of an interval. Preliminary definition of such a measure; properties it should satisfy. The non-existence of such a measure if the sets to be measured are left arbitrary. Review of basic topology. Definition of a $\sigma$-algebra. The $\sigma$-algebra generated by a subset $S$ of a set $X$ is the smallest $\sigma$-algebra $\mathfrak{M}$ such that $\mathfrak{M}$ contains $S$ (the intersection of all $\sigma$-algebras containing $S$). Definition of Borel sets. For a topological space $(X,\mathcal{T})$, the Borel sets are the elements of the $\sigma$-algebra generated by $\mathcal{T}$. Definition of a measurable space. Definition of a measurable function. A function is measurable iff the preimages of open sets are contained in the $\sigma$-algebra of the domain. - Jason L.
9/15Problem session
9/16
  • Borel maps. Compositions of measurable maps are measurable.
    Topological bases. Second countable spaces. A map from a measurable space to a second countable space is measurable iff preimages of basis elements are measurable. Sums, differences, products and complexification of real-valued measurable functions are measurable. For complex-valued measurable functions: modulus, real and imaginary parts, sums, differences and multiplications are also measurable. Topology on the extended real line $\overline{\mathbb{R}}=[-\infty,+\infty]$. - Victor C.
29/19
  • A function from a measurable space $(X,\mathfrak{M})$ to $\overline{\mathbb{R}}$ is measurable iff $\{x \in X\;:\; f(x)>a\}\in\mathfrak{M}$ for any $a\in\mathbb{R}$.
    (Pointwise) limits of measurable $\overline{\mathbb{R}}$-valued functions are measurable. Simple functions. Every (measurable) function with values in $[0,+\infty]$ is the limit of a sequence of non-negative (measurable) simple functions. - Pepper H.
9/21
  • Measure spaces. Properties of measures: finite additivity, monotonicity, etc.
    Examples: Dirac $\delta$-measure, counting measure. - Sarah M.
Lecture by Prof. Williams
9/23
  • Main difference between Riemann and Lebesgue integrals: intervals are replaced with measurable sets in Lebesgue. Definition of Lebesgue integral for measurable simple functions and measurable functions on a measure space. Lebesgue integrals over subspaces. Scaling measures using simple functions. Monotone convergence theorem. - Harsh D.
Lecture by Prof. van Erp
39/26
  • Series of non-negative measurable functions. Fatou's Lemma. Measure defined by a non-negative density. Complex-valued integrable functions, $\mathscr{L}^1(\mu)$. Real and imaginary parts, positive and negative parts of real-valued functions. Integral of complex-valued functions. Linearity. If $f\in\mathscr{L}^1(\mu)$, then $\left|\int f\,d\mu\right|\leq|\int |f|\,d\mu$
9/28
  • Lebesgue's Dominated Convergence Theorem. Measures are countably subadditive. Equality almost everywhere: $f=g$ a.e. $[\mu]$ if $\mu(\{x:f(x)\neq g(x)\})=0$. It is an equivalence relation and implies $\int f\,d\mu = \int g\,d\mu$. A measure space is complete if all subsets of null sets are measurable. Completion of an arbitrary measure space. In a complete measure space, if $f=g$ a.e. $[\mu]$ and $g$ is measurable then so is $f$. - Xingru C.
9/30
  • On a complete measure space, if $f$ is a.e. equal to the limit of a series of measurable functions, $f$ is measurable. If $f$ is non-negative and $\int f\,d\mu=0$ then $f=0$ a.e. $[\mu]$. For $f$ complex-valued, if $\int_E f\,d\mu=0$ for every measurable subset $E$, then $f=0$ a.e. $[\mu]$. Outer measures are monotonic and countably sub-additive. Carathéodory Theorem: $\sigma$-algebra and complete measure associated with an outer measure. Lebesgue measure on the real line. - Myles W.
Not in text
410/03
  • Existence and abundance of non-Lebesgue measurable sets. Cantor set, Cantor-Lebesgue function.
Not in text
10/05
  • The Borel $\sigma$-algebra on the real line is strictly included in the Lebesgue $\sigma$-algebra. Algebras and premeasures. Extension of premeasures. Uniqueness in the $\sigma$-finite case. Case of the real line. - Yunxiang W.
Folland
10/06Problem session
10/07
  • Premeasure on the algebra generated by rectangles. Measurability of slices. The Monotone Class Lemma.
Folland
510/10
  • Proof of the Monotone Class Lemma. Tonelli and Fubini.
Folland
10/12
  • Proof of Fubini's Theorem. Normed linear spaces, Cauchy sequences, Banach spaces. Characterization by absolutely convergent series. Equivalence almost everywhere in $\mathscr{L}^1(X,\mathfrak{M},\mu)$. The quotient $L^1(X,\mathfrak{M},\mu)$ is a Banach space.
10/13Problem sessionComparison of the Riemann and Lebesgue integrals
10/14
  • $\mathbb{C}$- and $\mathbb{R}$-valued measures. Positive and negative sets. Every set of positive measure contains a positive set of positive measure. Hahn decomposition of a measurable space equipped with a signed measure. Jordan decomposition of a signed measure.
610/17
  • Modes of convergence: pointwise, $\mu$-almost everywhere, uniform, almost uniform, $L^1$. Comparisons and counterexamples. Convergence in measure. It is implied by $L^1$-convergence. If a sequence of measurable functions converges in measure, it has a subsequence that converges almost everywhere.
10/19
  • If $\mu$ and $\nu$ are $\sigma$-finite measures on a measurable space $(X,\mathfrak{M})$ such that $\nu<<\mu$, then $\nu$ admits a density with respect to $\mu$. Two such densities coincide almost everywhere. Heuristics if $\mu$ is the Lebesgue measure on the real line. Proof.
10/20Midterm examinationElements of solution
10/21
  • Domains and regions in $\mathbb{C}$. Differentiability of functions of the complex variable. Holomorphy. Convergence of power series. Analytic functions are holomorphic. Uniqueness of expansions.
710/24
  • Construction of analytic functions. Curves in $\mathbb{C}$. Elementary properties of paths.
10/26
  • Definition and properties of the index of a path in $\mathbb{C}$. Examples.
10/27
  • The Fundamental Theorem for line integrals. Cauchy's Theorem for triangles.
10/28
  • Cauchy's Theorem for convex domains. Cauchy's Formula. Morera's Theorem. Limit points.
810/31
  • Orders of zeroes of holomorphic functions. Principle of isolated zeroes. Isolated singularities: sufficient conditions for singularities to be removable. Classification of isolated singularities.
11/02
  • Cauchy estimates on derivatives. Uniform convergence on compacts. It preserves holomorphy. Comparison with the case of functions of the real variable.
11/03Problem session
11/04
  • Counting zeros of a holomorphic function enclosed by a closed path. Local constance of the number of preimages under a holomorphic function. Open Mapping Theorem. Maximum principle. Local holomorphic inversion.
911/07
  • Chains are classes of formal sums of paths. Cycles. Paths that are homologous to zero. In a convex region, every closed path is homologous to zero.
11/09
  • Cauchy's Theorem for cycles in arbitrary domains. Cauchy's formula in annuli. Homotopy, simply connected spaces. Convex regions are simply connected. Invariance of the index under homotopy. Comparison of null-homotopic paths and paths homologous to zero.
11/10
  • Proof of the homotopy invariance theorem. Definition of meromorphic functions and residues. The Residue Theorem.
11/11
  • Existence and uniqueness of Laurent series expansions for meromorphic functions.
1011/14Wrap-up
11/15Problem sessionfrom 2:30pm in Kemeny 120
11/18Final examination Elements of solution

Mathematics Department ∈ Dartmouth College