Instructor: | Semyon Dyatlov |
Textbook: | Daniel Stroock, Essentials of Integration Theory
for Analysis [ Errata ] From an MIT computer, you can download the book or order a softcover edition for $25 here For the last two lectures, I will use Fraydoun Rezakhanlou's lecture notes Some movies about chaotic dynamics |
Class hours: | MWF 10–11 in 2–135 |
Office hours: | Mon 3–4, Tue 1:30–2:30 in 2–273 |
Course info: | click here |
Wed | May 11 |
Dynamical systems: invariant sets and ergodic
measures. The L^{1} and
L^{2} ergodic theorems.
Orthogonality of ergodic measures.
Mixing. Example: an expanding map on the circle.
Homework 13 due |
Mon | May 9 | Dynamical systems: ergodic averages and invariant measures. Existence of invariant measures. Invariant measures coming from closed orbits. Example: irrational shift on the circle. |
Fri | May 6 | §8.3. Hausdorff measures and Hausdorff dimension (§8.3.3). Examples, including hypersurfaces (Theorem 8.3.17, without proof). Outer measures and Carathéodory measurable sets (Theorems 8.3.4, 8.3.6, without proof). |
Wed | May 4 |
§8.2. Daniell integration theory; Stone's Theorem (Theorem 8.2.11,
without proof). Dini's Lemma (Lemma 8.2.15). Riesz Representation Theorem
for the space of continuous functions (Theorem 8.2.16).
Homework 12 due |
Mon | May 2 | §7.3,8.1. L^{2} theory of Fourier transform (very briefly, without using Hermite functions). Radon–Nikodym Theorem and Lebesgue's Decomposition Theorem for general measures (Lemma 8.1.2, Theorem 8.1.3). |
Fri | Apr 29 | §7.3. Basic properties of Fourier transform (Lemma 7.3.3). Fourier transform of Gaussians (Lemma 7.3.5). Fourier Inversion Formula (Theorem 7.3.9). |
Wed | Apr 27 |
§8.1,7.2. Riesz Representation Theorem for Hilbert spaces
(Theorem 8.1.1). Fourier series: completeness of Fourier basis (Theorem 7.2.2).
Homework 11 due |
Mon | Apr 25 | §7.1. Existence of orthonormal bases in Hilbert spaces (Lemma 7.1.5). Writing an element of a Hilbert space via an orthonormal basis (Lemma 7.1.6). |
Fri | Apr 22 | §6.3,7.1. Friedrichs mollification (Theorem 6.3.12, Corollary 6.3.13, Theorem 6.3.15). Hilbert spaces; orthogonal projection (Theorem 7.1.3). |
Wed | Apr 20 |
§6.3. Young's inequality (Theorem 6.3.5). Basic properties
of convolutions (Lemma 6.3.7, Theorem 6.3.9, Lemma 6.3.10).
Homework 10 due |
Fri | Apr 15 | §6.2,6.3. More properties of L^{p} spaces (Theorem 6.2.4). Mixed Lebesgue spaces and their properties (Lemma 6.2.5, Theorem 6.2.7). Integral kernels and L^{p} estimates (Lemma 6.3.1, Theorem 6.3.2). |
Wed | Apr 13 |
§6.2. Convergence in L^{p} spaces;
completeness and separability (Theorem 6.2.1).
Homework 9 due |
Mon | Apr 11 | §6.1,6.2. Minkowski's inequality (Theorem 6.1.4). L^{p} is a normed vector space. Hölder's inequality (Theorem 6.1.5) and applications. Dual spaces to L^{p} (without proof). |
Fri | Apr 8 | §6.1,6.2. Convex sets and concave functions; criterion for concavity for C^{2} functions (Lemma 6.1.2). Jensen's inequality (Theorem 6.1.1). |
Wed | Apr 6 |
§5.2,6.2. Finishing surface measure:
proof of Lemma 5.2.14. How to calculate
surface integral (Theorem 5.2.16).
L^{p} spaces: definition and
why they are useful.
Homework 8 due |
Mon | Apr 4 | §5.1,5.2. More about hypersurfaces: existence of coordinate charts (Lemma 5.2.10) and change of coordinates. Tangent spaces. Surface measure as the limit of volumes of tubular neighborhoods (see (5.2.9)). Surface measure in a coordinate chart (Lemma 5.2.14, to be proved Wednesday). Integrals over R^{N} written in spherical coordinates (Theorem 5.1.8). |
Fri | Apr 1 | §5.1,5.2. Pushforward of a measure and abstract change of variables (Lemma 5.1.1). A brief introduction to differential geometry: hypersurfaces in R^{N} (§5.2.2). Coordinate charts. Honorable mention of differential geometry concepts we will not cover: manifolds, integration of differential forms, general Stokes' theorem, volume forms on Riemannian manifolds |
Wed | Mar 30 |
§5.2. End of proof of Jacobi's formula (Theorem 5.2.2): estimating the measure
of (c) open set compactly contained in the domain
(d) general Lebesgue measurable set.
Inverse mapping
theorem (without proof).
Homework 7 due |
Mon | Mar 28 | §5.2. Jacobi's Formula (Theorem 5.2.2): estimating the Lebesgue measure of (a) the image of the unit square under a C^{1} transformation whose differential is close to the identity (b) the image of any square under a C^{1} transformation whose differential is close to the differential at the corner of the square. |
Fri | Mar 18 | §4.2. Steiner symmetrization and isodiametric inequality (Lemma 4.2.3, Theorem 4.2.4). N-dimensional Hausdorff measure on R^{N}. Open sets can be packed up with small balls up to a Lebesgue measure zero set (Lemma 4.2.6). Hausdorff measure coincides with Lebesgue measure (Theorem 4.2.7). |
Wed | Mar 16 |
§4.1,4.2. Proof of Tonelli's Theorem (Lemmas 4.1.1–4.1.3).
Isodiametric inequality: sketch of the proof.
Homework 6 due |
Mon | Mar 14 | §3.3,4.1. Proofs of Theorems 3.3.8 and 3.3.14. Products of measures. Tonelli's Theorem for finite measures (Lemma 4.1.3, to be proved Wednesday). General Tonelli's Theorem and Fubini's Theorem (Theorems 4.1.5, 4.1.6). |
Fri | Mar 11 | §3.3. Absolutely continuous functions are indefinite integrals (Lemma 3.3.5, Theorem 3.3.6). Lebesgue decomposition of measures (Theorem 3.3.12). Hardy–Littlewood Maximal Function. Indefinite integrals are almost everywhere differentiable (Theorem 3.3.8, to be proved Monday). Singular distribution functions have zero derivative almost everywhere (Theorem 3.3.14, to be proved Monday). Fundamental Theorem of Calculus for absolutely continuous functions. Lebesgue Differentiation Theorem (Theorem 3.3.15). |
Wed | Mar 9 |
§3.3. Definitions of absolutely continuous and singular measures.
Sunrise Lemma (Theorem 3.3.1). Controlling Lebesgue
measure of places where a distribution function is too steep (Corollaries 3.3.2, 3.3.3).
Homework 5 due |
Mon | Mar 7 | §3.2,3.3. Dense subsets of L^{1} (Theorem 3.2.14, Corollary 3.2.15). Starting structure theory of finite measures on R: Lipschitz continuous distribution functions can be written as indefinite integrals (Lemma 3.3.4). |
Fri | Mar 4 | §3.2. Convergence in measure implies a subsequence converges almost everywhere (Theorem 3.2.10). Fatou's Lemma and Dominated Convergence Theorem hold for convergence in measure (Theorem 3.2.12). L^{1} is a Banach space (Lemma 3.2.13). Motivation for completeness and density: continuous linear extension. |
Wed | Mar 2 |
§3.2. Fatou's Lemma and Dominated Convergence Theorem (Theorems 3.2.3–3.2.5).
Convergence in L^{1} implies convergence in measure;
convergence almost everywhere on finite measure space implies
convergence in measure (Theorem 3.2.7).
Homework 4 due |
Mon | Feb 29 | §3.1,3.2. The space L^{1} (§3.1.2). Pointwise limits of measurable functions are measurable (Lemma 3.2.1). Monotone Convergence Theorem (Theorem 3.2.2). |
Fri | Feb 26 | §3.1. Finishing Lebesgue integral construction: passing from simple measurable nonnegative functions to general measurable functions (Lemmas 3.1.4–3.1.8). |
Wed | Feb 24 |
§3.1. Continuing Lebesgue integral construction: properties
of measurable functions. The extended real line R. Simple functions and
their integrals (up to Lemma 3.1.2).
Homework 3 due |
Mon | Feb 22 | §2.2,3.1. Stieltjes and Bernouilli measures (§2.2.3–2.2.5, very briefly). Starting Lebesgue integral construction: measurable maps and functions. |
Fri | Feb 19 | §2.2. Finishing Lebesgue measure construction: countable additivity, regularity, and completeness (Lemma 2.2.9, Theorem 2.2.10). Measure zero sets and `almost everywhere'. Distortion of Lebesgue measure under translations and linear maps (Corollary 2.2.14, Theorem 2.2.15). |
Wed | Feb 17 |
§2.2. Continuing Lebesgue measure construction: measurable
sets form a σ-algebra (Lemmas 2.2.7–2.2.9).
Homework 2 due |
Tue | Feb 16 | §2.2. Starting Lebesgue measure construction: exterior Lebesgue measure and its properties (Lemmas 2.2.2–2.2.5). |
Fri | Feb 12 | §2.1,2.2. Finishing uniqueness of Lebesgue measure. Completion and regularity of measures (Lemma 2.1.14, Theorem 2.1.15). |
Wed | Feb 10 |
§2.1,2.2. Breaking open subsets of R^{n} into squares (Lemma 2.2.12). Uniqueness of Lebesgue measure (Theorem 2.2.13); Π- and Λ-systems
(Lemma 2.1.12, Theorem 2.1.13).
Looking forward to Lebesgue measure construction!
Homework 1 due |
Mon | Feb 8 | §2.1. Definition of σ-algebra, measure, and measurable maps; basic properties. Counting measure. Borel σ-algebra; continuous maps are Borel measurable. Why can't all sets be measurable? |
Fri | Feb 5 | §1.1,1.2. A very brief review of Riemann integration. |
Wed | Feb 3 | Introduction: what measure and integration are supposed to mean. Examples. |