note-math

constant function ${𝑓}_{𝑖}$ support on simplex ${𝜎}_{𝑖}$ + $𝑓=\sum {𝑓}_{𝑖}{𝜎}_{𝑖}$ countable infinite combination + ${\int }_{𝜎}𝑓=\sum {𝑓}_{𝑖}\left({𝜎}_{𝑖}\right)\text{Vol}\left({𝜎}_{𝑖}\right)$ absolutely convergent

Or use continuous piecewise affine linear or simplicial function e.g. for simplex with vertices ${𝑥}_0,\dots ,{𝑥}_{𝑛}$, $𝑓\left(\sum {𝑡}_{𝑖}{𝑥}_{𝑖}\right)={𝑡}_{𝑖}𝑓\left({𝑥}_{𝑖}\right)$. The integral is defined as the average of the values according to the vertices i.e. the result of the affine center mapping $𝑓\left(\frac{{𝑥}_0+\cdots +{𝑥}_{𝑛}}{𝑛+1}\right)\text{Vol}\left(\text{simp}\left({𝑥}_1,\dots ,{𝑥}_{𝑛}\right)\right)$

$\mathrm{\varphi }\in {𝐿}^1$ := $\forall 𝜀>0$ there exist piecewise constant functions $𝑓,𝑔$ such that $|\mathrm{\varphi }-𝑓|\le \left|𝑔\right|$ and $\Vert 𝑔\Vert =\int \left|𝑔\right|<𝜀$

Similarly for ${𝐿}^2$

In the piecewise constant space, similar to the definition of measurable sets, for integral distance, triangle inequality, the limit is unique

So that we can define $𝑓\to \mathrm{\varphi }$ and $\int \mathrm{\varphi }=\underset{𝑓\to \mathrm{\varphi }}{\text{ lim }}\int 𝑓$

$\mathrm{\varphi }\in {𝐿}^1\left({\mathrm{ℝ}}^{𝑑},{\mathrm{ℝ}}^{{𝑑}^{\prime }}\right)⟺\left|\mathrm{\varphi }\right|\in {𝐿}^1\left({\mathrm{ℝ}}^{𝑑},\mathrm{ℝ}\right)$

$\mathrm{\varphi }\in {𝐿}^2\left({\mathrm{ℝ}}^{𝑑},{\mathrm{ℝ}}^{{𝑑}^{\prime }}\right)⟺\left|\mathrm{\varphi }\right|\in {𝐿}^2\left({\mathrm{ℝ}}^{𝑑},\mathrm{ℝ}\right)⟺|\mathrm{\varphi }{|}^2\in {𝐿}^1\left({\mathrm{ℝ}}^{𝑑},\mathrm{ℝ}\right)$

The bisection walk sequence is Cauchy in integral distance, and the measure tends to $0$

$$\begin{array}{ccc}{𝑓}_1& =& {\mathrm{𝟙}}_{\left[0,\frac{1}{2}\right]}\\ {𝑓}_2& =& {\mathrm{𝟙}}_{\left[\frac{1}{2},1\right]}\\ {𝑓}_3& =& {\mathrm{𝟙}}_{\left[0,\frac{1}{4}\right]}\\ & \cdots \\ {𝑓}_{2+\cdots +2^{𝑘}+𝑝}& =& {\mathrm{𝟙}}_{\left[\frac{𝑝-1}{2^{𝑘}},\frac{𝑝}{2^{𝑘}}\right]}\end{array}$$

It does not follow the definition of pointwise convergence

$$\begin{array}{c}\forall 𝑥\in \left[0,1\right],\forall 𝑁\in \mathrm{\mathbb{N}},\exists 𝑖,𝑗>𝑁\\ {𝑓}_{𝑖}\left(𝑥\right)=0,{𝑓}_{𝑗}\left(𝑥\right)=1\end{array}$$

Although conceptually it converges to the empty set

But for all ${𝐿}^1,{𝐿}^2$ integral distances, there exists a subnet in the Cauchy net that converges almost everywhere pointwise to the target integrable function. This comes from the fact that there exists a set $𝐴$ with arbitrarily small measure such that it converges absolutely and uniformly on ${𝐴}^{\complement }$

let $𝑓$ almost everywhere analytic

The change of variable formula formula for diffeomorphisms of integrals is ${\int }_{{\mathrm{ℝ}}^{𝑛}}𝑓={\int }_{{\mathrm{ℝ}}^{𝑛}}\left(𝑓\circ \mathrm{\varphi }\right)|\text{det }𝑑\mathrm{\varphi }|$ or ${\int }_{{\mathrm{ℝ}}^{𝑛}}𝑑𝑦𝑓\left(𝑦\right)={\int }_{{\mathrm{ℝ}}^{𝑛}}𝑑𝑥\left(𝑓\circ \mathrm{\varphi }\right)\left(𝑥\right)|\text{det }𝑑\mathrm{\varphi }\left(𝑥\right)|$

The differential of the coordinate transformation map $𝑑𝑓$ at each simplex center as an affine map acting on the domain simplex is used to obtain the range space simplex for approximation, then use (high order) #link(<mean-value-theorem-analytic>)[], then take partition limit (ref-12, p.92–99)

It is necessary to first perform compact uniform control on the bounded region for the approximation of the differential mean value theorem

Then the unbounded region is a countable approximation from the bounded region, using the ${\sum }_{𝑖=1..\infty }{𝜀}_{𝑖}<𝜀$ technique

If it is considered as integrating an $𝑛$ form, (cf. #link(<integral-on-form>)[]) then the integral change of variable formula is equivalent to $𝑛$ form integration is diffeomorphism invariant

According to the change of variable formula, the integral of $𝑛$ form in the coordinates on the manifold is invariant (cf. #link(<integral-on-form>)[])

But what if we want to integrate the $𝑛$ form defined on the entire #link(<orientable>)[] manifold?

One way is, similar to the proof of #link(<integral-change-of-variable-formula>)[], in coordinate, linear approximation + compact uniformly control + partition limit, then use countable cover to approximate entire manifold

In order to define the integral, some kind of countability assumption is needed. The simplest assumption is that the manifold can be covered by countable coordinate cards. Let's use this assumption

Now the problem is that the integral at the intersection of the coordinate cards is repeated and needs to be removed

I will not use measurable sets being closed under intersection and set-minus, nor use curved simplex (box) type region partition alias triangulation, which is even more difficult to proof

Instead, I will use the linear approximation version of triangulation. Under linear approximation, simplex is closed under intersection and set-minus (polyhedra decompose to to simplex)

e.g. the differential $𝑑𝑓$ of the transition map at each simplex (box) center as an #link(<affine-map-point-ver>)[] (linear-map) to transform the simplex (box) to the simplex of the coordinate region $𝐴$. Then the intersection and reduction of the simplex (box) can be decomposed into the simplex (box) again.

Taking the limit of this approximation, require ${𝐿}^1,{𝐿}^2$ style absolute convergence, gives the integral on the manifold

Prove that the result does not depend on the choice of coordinate system and linear approximation method

The integral of a form over a $𝑘$-simplex is also invariant, and there's no need to define volume for lower-dimensional simplices.

constant-type form

simplicial map type $𝑘$ form := Let ${𝑥}_0,\dots ,{𝑥}_{𝑘}$ be the vertices of the simplex, then $𝜔\left(\sum {𝑡}_{𝑖}{𝑥}_{𝑖}\right)=\sum {𝑡}_{𝑖}𝜔\left({𝑥}_{𝑖}\right)$

Similar to the $𝑛$-order case, the integral of a simplicial map form on a $𝑘$-chain is defined as ${\int }_{𝜎}𝜔=\sum 𝜔\left(\text{center of }{𝜎}_{𝑖}\right)\text{Vol}\left({𝜎}_{𝑖}\right)$

Even if two $𝑘$-simplices are adjacent, their orientations might be discontinuous. This is different from the $𝑛$-order case, where the codimension is zero, so all $𝑛$-simplices have the same orientation.

Two adjacent $𝑘$-simplices share common vertices, and the form acting on these points might have different values. The integral of a simplicial type form is also equivalent to taking the average of the directions at the vertices.

A good approximation would require the $𝑘$-directions to have good regularity, but without additional structure, it seems difficult to define such a concept (even for a Grassmann manifold?).

submanifold structure can simply eliminate this $𝑘$-direction discontinuity.

let $𝑀$ be $𝑛$ dimension manifold, $𝑘$ form restrict to #link(<orientable>)[] $𝑘$ submanifold tangent space

$𝑛$ form in $𝑀$ is equivalent to scalar function, but how to control $𝑛-1$ form by integral? try $\sup \left(𝑆:\text{ orientable }𝑛-1\text{ submanifold}\right)\left({\int }_{𝑆}|𝜔-{𝜔}^{\prime }|\right)$?

if unnecessary, do not introduce metric to define ${\int }_{𝑀}{\left({⟨𝜔⟩}^2\right)}^{\frac{1}{2}}$ or ${\left({\int }_{𝑀}{⟨𝜔⟩}^2\right)}^{\frac{1}{2}}$ now