# measure theory (blogs) — RSS Amplifier Recent posts from the 4 feeds in the RSS Amplifier directory that cover measure theory. Page: Feed: --- ## [On the theorem of Gelfand-Mazur](https://freedommathdance.blogspot.com/2026/04/on-theorem-of-gelfand-mazur.html) _2026-04-13 · Antoine Chambert-Loir · Freedom Math Dance_ In its classical statement, the theorem of Gelfand-Mazur is the following: Theorem 1 . — Let $K$ be a complete normed field which is an extension of the field $\\mathbf C$ of complex numbers. Then $K=\\mathbf C$. As in the history of any theorem, the reality is a bit more complicated. The theorem proved by Stanislas Mazur in 1938 concerns all fields $F$, extensions of the field $\\mathbf R$ of real… ## [Joyal's proof of Cayley's formula for the enumeration of trees](https://freedommathdance.blogspot.com/2026/02/joyals-proof-of-cayleys-formula-for.html) _2026-02-18 · Antoine Chambert-Loir · Freedom Math Dance_ That formula of Cayley asserts that there are $n^{n-2}$ trees with vertices labeled $1,\\dots,n$. This is a chapter in graph theory, in which graphs are “simple”. More precisely, let's call a graph structure on a set $V$ (called “vertices”) is the datum of a subset $E$ of 2-element subsets of~$V$, called “edges”. So an edge is of the form $\\{a,b\\}$, where $a$ and $b$ are distinct elements of $V$,… ## [Associated prime ideals and regular elements in polynomial rings](https://freedommathdance.blogspot.com/2026/02/associated-prime-ideals-and-regular.html) _2026-02-13 · Antoine Chambert-Loir · Freedom Math Dance_ This post is here to record a fact that ought to be better known and for which I ought to have known a proper proof beforehand. Let's start by defining two thirds of the terms of the title. By polynomial ring , I simply mean a ring of polynomials $A\[T\_1,T\_2,\\dots\]$ over a commutative ring $A$, in any number of indeterminates. The most important case here will however be that of a ring of… ## [The determinant of transvections (an update)](https://freedommathdance.blogspot.com/2025/11/the-determinant-of-transvections-update.html) _2025-11-30 · Antoine Chambert-Loir · Freedom Math Dance_ In the previous post , I had explained how I could prove a general version of the classic fact that transvections have determinant 1. Recall here that transvections in an $R$-module $M$ are linear maps of the form $x\\mapsto x+f(x)v$, where $f$ is a linear form on $M$ and $v\\in M$ is a vector such that $f(v)=0$. To be able to talk of determinant, I assumed that $M$ had a finite basis, so that… ## [The determinant of transvections](https://freedommathdance.blogspot.com/2025/11/the-determinant-of-transvections.html) _2025-11-06 · Antoine Chambert-Loir · Freedom Math Dance_ A transvection in a $K$-vector space $V$ is a linear map $T(f,v)$ of the form $x\\mapsto x + f(x) v$, where $f\\in V^\*$ is a linear form and $v\\in V$ is a vector such that $f(v)=0$. It is known that such a linear map is invertible, with inverse given by $f$ and $-v$. More precisely, one has $T(f,0)=\\mathrm{id}$ and $T(f,v+w)=T(f,v)\\circ T(f,w)$. In finite dimension, these maps have determinant $1$… ## [The two adjunctions of the preimage](https://freedommathdance.blogspot.com/2025/09/the-two-adjunctions-of-preimage.html) _2025-09-06 · Antoine Chambert-Loir · Freedom Math Dance_ Sometimes in mathematics, you are told about very elementary things of which you hadn't even thought. I was well aware of some “duality” between image and preimage, but I just learned from Anatole Dedecker (who learned it from Patrick Massot) about another “duality” between preimage and some other notion. Moreover, it appears that this new notion can be used for making slightly more natural a… ## [The Krull dimension of the semiring of natural numbers is equal to 2](https://freedommathdance.blogspot.com/2025/07/the-krull-dimension-of-natural-numbers.html) _2025-07-14 · Antoine Chambert-Loir · Freedom Math Dance_ Let $R$ be a ring. Its Krull dimension is the supremum of the lengths $n$ of chains $P\_0\\subsetneq P\_1 \\subsetneq\\dots\\subsetneq P\_n$ of prime ideals of $R$. When $R$ is a field, the null ideal is the only prime ideal, and it is a maximal ideal so that its Krull dimension is zero. When $R$ is a principal ideal domain which is not a field, there are two kinds of prime ideals: the null ideal is… ## [Autoformalization of mathematical theorems? No shit!](https://freedommathdance.blogspot.com/2025/07/autoformalization-of-mathematical.html) _2025-07-02 · Antoine Chambert-Loir · Freedom Math Dance_ I've been formalizing mathematical theorems in Lean for some years now, and one of the major blocks is the difficulty of formalizing elementary results that mathematicians do not take the time to even state. For example, a mathematician's integer can implicitly be a natural number at a line and a real number at the next one, while proof assistants require that some “coercion” maps be introduced.… ## [Yet another proof of the Weierstrass approximation theorem](https://freedommathdance.blogspot.com/2025/04/yet-another-proof-of-weierstrass.html) _2025-04-25 · Antoine Chambert-Loir · Freedom Math Dance_ Browsing through my Zotero database, I fall upon a paper by Harald Kuhn where he proposes an elementary proof of the Weierstrass approximation theorem. The proof is indeed neat, so here it is. Theorem. — Let $f\\colon\[0;1\]\\to\\mathbf R$ be a continuous function and let $\\varepsilon$ be a strictly positive real number. There exists a polynomial $P\\in\\mathbf R\[T\]$ such that… ## [Yet another proof of the inequality between the arithmetic and the geometric means](https://freedommathdance.blogspot.com/2025/04/yet-another-proof-of-inequality-between.html) _2025-04-24 · Antoine Chambert-Loir · Freedom Math Dance_ This is an exposition of the proof of the inequality between arithmetic and geometric means given by A. Pełczyński (1992), “ Yet another proof of the inequality between the means ”, Annales Societatis Mathematicae Polonae. Seria II. Wiadomości Matematyczne , 29 , p. 223–224. The proof might look bizarre, but I can guess some relation with another paper of the author where he proves uniqueness of… ## [Transcript-Driven Stock Discovery (Sponsored)](https://crawlproof.com/a/fJJB1EQROKbh) _2025-04-24 · **Sponsored**_ Ranked, evidence-backed watchlists from indexed executive transcripts for 1–2 quarter horizons ## [A generalization of the Eisenstein criterion](https://freedommathdance.blogspot.com/2025/04/a-generalization-of-eisenstein-criterion.html) _2025-04-06 · Antoine Chambert-Loir · Freedom Math Dance_ Recently, in the Zulip server for Lean users, somebody went with something that looked like homework, but managed to sting me a little bit. It was about irreducibility of polynomials with integer coefficients. Specifically, the guy wanted a proof that the polynomial $T^4-10 T^2+1$ is irreducible, claiming that the Eisenstein criterion was not good at it. What was to proven (this is what… ## [A simple proof of a theorem of Kronecker](https://freedommathdance.blogspot.com/2025/03/a-simple-proof-of-theorem-of-kronecker.html) _2025-03-29 · Antoine Chambert-Loir · Freedom Math Dance_ Kronecker's theorem of the title is the following. Theorem. — Let $\\alpha\\in\\mathbf C$ be an algebraic integer all of whose conjugates have absolute value at most $1$. Then either $\\alpha=0$, or $\\alpha$ is a root of unity. This theorem has several elementary proofs. In this post, I explain the simple proof proposed by Gebhart Greiter in his American Mathematical Monthly note , adding details so… ## [Implementing a Maximal-Munch Lexical Analyzer](https://abhinavpradeep.com/blog/compiler-2/) _2025-01-30 · Abhinav Pradeep's Blog_ Extending the NFA generator Previously we implemented a regular expression matching algorithm that generated an NFA from an input regular expression, and simulated a string over the NFA to check for acceptance. The regular expression was required to be in postfix form and the algorithm only supported union $|$ , concatenation $.$ , Kleene star $\*$ , one-or-more $+$ , and zero-or-one $?$… ## [Regular expression matching: Theory and Implementation](https://abhinavpradeep.com/blog/compiler-1/) _2025-01-25 · Abhinav Pradeep's Blog_ Introduction The first step in building a compiler would be to build a lexical analyzer, shortened to ’lexer’. To define what a lexer does we may understand it as a function: $$\\text{Lexer :: string} \\to \[\\text{token}\]$$ Where a token can be thought of as the tuple $\\text{token} = \\langle \\text{class, string} \\rangle$. A token class, loosely speaking, corresponds to a set of strings.… ## [On numbers and unicorns](https://freedommathdance.blogspot.com/2025/01/on-numbers-and-unicorns.html) _2025-01-11 · Antoine Chambert-Loir · Freedom Math Dance_ Reading a book on philosophy of mathematics, even if it's written lightly, such as that one, Why is there philosophy of mathematics at all? by Ian Hacking, may have unexpected effects. The most visible one has been a poll that I submitted on Mastodon on December 4th. As you can read, there were four options: Numbers exist Unicorns exist Numbers have more existence than unicorns Neither numbers no… ## [ML Notes 2025. Supervised Learning and the Problem of Induction](http://jyotirmoy.net/posts/2024-12-30-ML%20Notes%202025.%20Supervised%20Learning%20and%20the%20Problem%20of%20Induction.html) _2024-12-30 · Jyotirmoy Bhattacharya: Blog_ Dec. 30, 2024 Supervised Learning Supervised learning is a fundamental type of machine learning where we learn from examples that have been labeled with their correct answers. The term “supervised” comes from the idea that some knowledgeable entity (the supervisor) has already gone through the data and provided the right answer for each example. This supervisor could be: Human experts manually… ## [The combinatorial Nullstellensatz](https://freedommathdance.blogspot.com/2024/09/the-combinatorial-nullstellensatz.html) _2024-09-12 · Antoine Chambert-Loir · Freedom Math Dance_ The “combinatorial Nullstellensatz” is a relatively elementary statement due to Noga Alon (1999) whose name, while possibly frightening, really says what it is and what it is good for. (A freely available version is there .) Nullstellensatz is the classic name for a theorem of David Hilbert that relates loci in $F^n$ defined by polynomial equations and the ideals of the polynomial ring…