18.125, Measure Theory and Analysis, Spring 2016

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 L1 and L2 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. L2 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 Lp spaces (Theorem 6.2.4). Mixed Lebesgue spaces and their properties (Lemma 6.2.5, Theorem 6.2.7). Integral kernels and Lp estimates (Lemma 6.3.1, Theorem 6.3.2).
Wed Apr 13 §6.2. Convergence in Lp 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). Lp is a normed vector space. Hölder's inequality (Theorem 6.1.5) and applications. Dual spaces to Lp (without proof).
Fri Apr 8 §6.1,6.2. Convex sets and concave functions; criterion for concavity for C2 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). Lp 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 RN 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 RN (§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 C1 transformation whose differential is close to the identity (b) the image of any square under a C1 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 RN. 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 L1 (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). L1 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 L1 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 L1 (§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 Rn 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.

Last updated: May 17, 2016