Math · 12
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Show that any countable subset of \(\mathbb{R}\) has Lebesgue outer measure zero.
Timer not started. Twenty minutes of real work before a solution.
Enumerate the set as \(x_n\). Cover \(x_n\) by an interval of length \(\varepsilon/2^n\). The total length is \(\varepsilon\). Outer measure is the infimum of such covers, so it is 0.
Attack 2
Why is the pointwise limit of the functions \(f_n=n\mathbf{1}_{(0,1/n)}\) on \([0,1]\) not something the Riemann integral handles gracefully? Compute the integrals and the limit.
Timer not started. Twenty minutes of real work before a solution.
Each \(f_n\) integrates to 1. The pointwise limit is 0 everywhere (at 0 the functions are 0 if you define them that way, and for \(x>0\) eventually \(1/n
Attack 3
State dominated convergence in one sentence, including the dominant.
Timer not started. Twenty minutes of real work before a solution.
If \(|f_n|\le g\) almost everywhere, \(g\) integrable, and \(f_n\to f\) almost everywhere, then \(f\) is integrable and the integrals converge to the integral of \(f\).
Measure theory replaces the Riemann integral, which cannot handle a pointwise limit of functions without an extra uniform-convergence hypothesis you often do not have. A measure assigns sizes to a σ-algebra of sets. Lebesgue measure on \(\mathbb{R}\) extends length. The integral is built from simple functions up, and the convergence theorems — monotone convergence, Fatou, dominated convergence — are the reason the subject exists. \(L^p\) spaces are complete. That completeness is what makes Fourier analysis and probability sit on the same foundation.
Sheldon Axler’s Measure, Integration & Real Analysis is free from the author and is the main text. Rudin’s Real and Complex Analysis is the paid ceiling; no ISBN was verified for that specific title in this pass, so none is printed. Folland is the other standard paid book, same situation. You do not need either to learn the convergence theorems. You need to prove them.
A set of measure zero is not a set you can ignore inside a definition. It is a set you can ignore inside a statement that has already been proved to depend only on almost-everywhere behavior.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Measures. A σ-algebra is closed under complements and countable unions. A measure is a countably additive \([0,\infty]\)-valued function on it, with \(\mu(\emptyset)=0\). Lebesgue measure is the completion of the restriction of outer measure to the Carathéodory-measurable sets, and it extends length on intervals.
Countable additivity for disjoint intervals is a standard estimate. Carathéodory’s criterion produces a σ-algebra on which outer measure is a measure. Completion adds subsets of null sets.
Integral. The integral of a nonnegative simple function is the obvious sum. For a nonnegative measurable function it is the supremum of integrals of simple functions underneath. For a general measurable function it is the difference of the positive and negative parts, when at least one is finite.
Well-definedness of the simple-function integral uses a common refinement of the level sets. Monotone convergence is then used to prove linearity for nonnegative functions. Write that once.
Convergence theorems. Monotone convergence: if \(0\le f_n\uparrow f\) then the integrals rise to the integral of \(f\). Fatou: the integral of the liminf is at most the liminf of the integrals. Dominated convergence is the statement in the attack.
Monotone convergence is the technical heart; Fatou is monotone convergence applied to the infima from each index onward; dominated convergence is Fatou applied to \(g+f_n\) and \(g-f_n\).
Fubini. For σ-finite measures, the integral of a nonnegative measurable function on a product equals the iterated integrals in either order. For integrable functions of either sign, the same holds, and almost every slice is integrable.
Prove it for indicators of measurable rectangles, then for simple functions, then by monotone convergence. The signed case uses integrability to justify the difference.
L^p. \(L^p\) is the space of measurable functions with \(\int |f|^p<\infty\), modulo almost-everywhere equality. It is a Banach space for \(1\le p\le\infty\). Hölder’s inequality is the tool that makes the dual of \(L^p\) look like \(L^q\).
Completeness is: a fast Cauchy sequence has a subsequence with \(\|f_{n_{k+1}}-f_{n_k}\|<2^{-k}\), the telescoping series converges absolutely almost everywhere by monotone convergence, and the limit is the \(L^p\) limit. Hölder is Young’s inequality integrated.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Axler, the measure-theory book site
Read once, do not study from this. Confirm the PDF is the free book and see the table of contents. Then work in the PDF.
The thing you work through.
Axler, Measure, Integration & Real Analysis
Free legal PDF from the author. This is the main text. It goes from measure through functional analysis at a level you can actually finish.
The problem book.
Axler’s exercises on the convergence theorems and L^p
The book is the problem set. Do not skip the “prove that this hypothesis is necessary” exercises.
The hardest questions that exist at this level.
A qualifying-exam style reconstruction of dominated convergence
Rudin’s Real and Complex Analysis and Folland are the paid ceilings; ISBNs were not verified, so they are not numbered here. The public ceiling is: write the three convergence theorems and a counterexample for each missing hypothesis, from memory.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: from memory, the proofs of monotone convergence’s consequences (Fatou and dominated convergence, assuming monotone convergence) and the construction of a non-integrable pointwise limit of integrable functions. Plus the completeness of \(L^1\). Fail if “almost everywhere” is not in the statements.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Define a measure, state dominated convergence with the dominant, and explain the spike counterexample. No notes.