Cédric #Villani, #Fields #Medal 2010 , On #Measure #Theory and #Lebesgue #Integration

https://youtube.com/watch?v=qBbq4RepjxE&si=nKL5OOYMsXEgIO1W

Cédric #Villani, #Fields #Medal 2010 , On #Measure #Theory and #Lebesgue #Integration youtube.com/watch?v=qBbq...

Cédric Villani - Fields Medal ...

DOMINATED CONVERGENCE THEOREM
Lebesgue's dominated convergence theorem provides sufficient conditions under which pointwise convergence of a sequence of functions implies convergence of the integrals. It's one of the reasons that makes #Lebesgue integration more powerful than #Riemann integration. The theorem an be stated as follows:

Let \((f_n)\) be a sequence of measurable functions on a measure space \((\mathcal{S},\Sigma,\mu)\). Suppose that \((f_n)\) converges pointwise to a function \(f\) and is dominated by some Lebesgue integrable function \(g\), i.e. \(|f_n(x)|\leq g(x)\ \forall n\) and \(\forall x\in\mathcal{S}\). Then, \(f\) is Lebesgue integrable, and

\[\displaystyle\lim_{n\to\infty}\int_\mathcal{S}f_n\ \mathrm{d}\mu=\int_\mathcal{S}f\ \mathrm{d}\mu\]
#ConvergenceTheorem #Convergence #DominatedConvergenceTheorem #Lebesgue #MeasurableFunction #LebesgueFunction #LebesgueIntegration #RiemannIntegration #MeasureSpace

`It is also called the #Cantor ternary #function, the #Lebesgue function, Lebesgue's singular function..the Devil's staircase, the Cantor #staircase function, and the Cantor–Lebesgue function. Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental #theorem of #calculus claimed by Harnack.`

https://en.wikipedia.org/wiki/Cantor_function

#devilsStaircase #fractal #cantorSet #topology #analysis

`In #mathematics, the Lebesgue differentiation #theorem is a theorem of real #analysis, which states that for almost every point, the value of an #integrable #function is the limit of #infinitesimal averages taken about the point. The theorem is named for Henri #Lebesgue. `

https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem

"For a probability p distribution in Rn with a p density f, such as equidistribution in an n-d ball wrt #Lebesgue measure => n randomly-independently chosen vectors will form a basis w p=1 as for n linearly dependent vectors x1, …, xn in Rn , det[x1 ⋯ xn] = 0

- It is consistent with ZF + DC that every set of reals is #Lebesgue measurable
due to #Solovay it cannot be proved in #ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or AD

- any σ-algebra generated by a countable collection of sets is separable, converse need not hold.
#Lebesgue L σ-algebra is separable (L measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).

#Hilbert Space, #lebesgue measure * and #functionalanalysis in general.
* yes it has more uses than just solving harder integrals

Unlike Fourier integrable f - #Riemann–#Lebesgue lemma fails for #measures

The #Riemann–#Lebesgue lemma states that the integral of a function like pfa is small. The integral will approach zero as the number of oscillations increases
there is almost always a way to tweet things wo #latex

All metric spaces have #Hausdorff completions c, let L1 be C of normed vector space c_c. C_c is #isomorphic to space of #Lebesgue integrable functions #modulo subspace of functions w integral 0. Riemann
∫ is a uniformly continuous functional wrt norm on Cc, dense in L1. ∫

All metric spaces have #Hausdorff completions c, let L1 be C of normed vector space c_c. C_c is #isomorphic to space of #Lebesgue integrable functions #modulo subspace of functions w integral 0. Riemann
∫ is a uniformly continuous functional wrt norm on Cc, dense in L1. ∫

there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional #Lebesgue measure, denoted dx.

there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional #Lebesgue measure, denoted dx.

linear combinations of translations of f are dense if and only if the zero set of the Fourier transform of f is empty (in the case of L1) or of #Lebesgue measure zero (in case of L2).

linear combinations of translations of f are dense if and only if the zero set of the Fourier transform of f is empty (in the case of L1) or of #Lebesgue measure zero (in case of L2).

When g(x) = x for all real x, then μg is the Lebesgue measure, and the Lebesgue–Stieltjes integral of  f  with respect to g is equivalent to the #Lebesgue integral of  f

When g(x) = x for all real x, then μg is the Lebesgue measure, and the Lebesgue–Stieltjes integral of  f  with respect to g is equivalent to the #Lebesgue integral of  f 

#Lebesgue integral classically requires the initial development of a workable measure theory before any useful results for the integral can be obtained