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Measure Theory(1)
Chapter 1: Motivation for Measure Theory As the first chapter of this series, we begin by presenting the motivation behind measure theory. Motivation Measure theory was developed to address the limitations of Riemann integration. Let’s first recall Riemann integration, which approximates the integral through upper and lower bounds, taking the limit as the partition becomes infinitely fine. However, this approach fails for certain special functions. For example: No…
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Hensel’s Lemma
We start from a problem: Problem: Find the number of elements in such that it is congruent to module for some integer . Solution. The problem is the same as finding the number of solutions to Using Chinese Remainder Theorem, the problem reduces to the system of equations Our goal, thus, is to find the number of solutions for each of the equation. We…
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A Problem on Outer Measure
There is a problem in Real Analysis by Halsey L. Royden, Patrick M. Fitzpatrick. Suppose that outer-measure is defined by covering sets by countable collections of closed, bounded intervals rather than coverings by open, bounded intervals. Show that the outer- measure remains unchanged. I give my answer here. Proof. Let be a set of real numbers. , defined similarly with intervals to be closed. Now consider , we form the…
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Some thoughts on proving Fourier Inversion(Complex analysis method)
In complex analysis, we have the following well-known theorem: Theorem If , then the Fourier inversion holds, namely, given We have The proof is not easy for a beginner who just touched this theorem. As for myself, I spent a tons of time on managing the proof. So I would like to share the logical thoughts on proving this theorem. It is intuitive to plug in…
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Harmonic Number(2)
Our goal is to prove the following theorem by Carlo Sanna. The proof is adapted and changed slightly from his paper. Theorem. For any prime number and any , we have Lemma 1 If is prime, , and , then and Proof. . Each of the sum is by Wolstenholme’s theorem. The second equation holds since Lemma 2 If is an odd…
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Harmonic Number (Starting from an AMC problem) (1)
The -th Harmonic number, denoted is defined by the sum of the reciprocals of the first natural numbers: This number has quite interesting properties. On thing we could consider when is finite is the quotient of the sum. If we write it as Then many problems raise. Here, we have a problem concerning when is the denominator the lcm of . 2022 AMC 12A Problem 23 Let…
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An inequality problem
Let be a continuous integrable real function on , then Solution 1 By Cauchy-Schwarz inequality, Solution 2 By Jensen’s inequality, since is a convex function, we have So Solution 3 By Hölder’s Inequality, we get Choose , we get the case of solution 1. We can also take and , which would be similar. Solution 4
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Measure Theory(1)
Chapter 1: Motivation for Measure Theory As the first chapter of this series, we begin by presenting the motivation behind measure theory. Motivation Measure theory was developed to address the limitations of Riemann integration. Let’s first recall Riemann integration, which approximates the integral through upper and lower bounds, taking the limit as the partition becomes…