[integral-piecewise-constant-function]
[integral-simplicial-function]
According to the decomposition of simplex intersection and subtraction, the finite addition and subtraction of piecewise constant functions is still a piecewise constant function
Define the integral distance . Or use norm
[Lebesgue-integrable]
:= there exist piecewise constant functions such that and
Similarly for
In the piecewise constant space, similar to the definition of measurable sets, for integral distance, triangle inequality, the limit is unique
[Lebesgue-integral]
So that we can define and
Example But it should be noted that, although the integral distance Cauchy net is always integrally convergent, there exists an integral distance Cauchy net that does not converge pointwise to the limit function
The bisection walk sequence is Cauchy in integral distance, and the measure tends to
It does not follow the definition of pointwise convergence
Although conceptually it converges to the empty set
[integrable-exist-subnet-almost-everywhere-pointwise-convergence] (ref-5, p.129โ130)
The measurable set defined by L^1,L^2 is a Lebesgue measurable set, which may be disconnected
What we define is absolutely integrable. Other integral operations, such as , are special limit operations based on absolute integrability, and are related to the environment of the problem
The linear change of coordinates gives the integral change of variable formula formula
[integral-on-form] For the integral over an -region, integrating a function over a volume is equivalent to integrating an -form. If considered as an integral of an -form, then the integral is invariant.
[integral-change-of-variable-formula]
let almost everywhere analytic
The change of variable formula formula for diffeomorphisms of integrals is or
The differential of the coordinate transformation map 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) mean-value-theorem-analytic, then take partition limit (ref-12, p.92โ99)
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 technique
If it is considered as integrating an form, (cf. integral-on-form) then the integral change of variable formula is equivalent to form integration is diffeomorphism invariant
[integral-on-manfold] Question
According to the change of variable formula, the integral of form in the coordinates on the manifold is invariant (cf. integral-on-form)
But what if we want to integrate the form defined on the entire orientable manifold?
One way is, similar to the proof of 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 of the transition map at each simplex (box) center as an affine-map-point-ver (linear-map) to transform the simplex (box) to the simplex of the coordinate region . Then the intersection and reduction of the simplex (box) can be decomposed into the simplex (box) again.
Taking the limit of this approximation, require 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 measures and integrals defined by rectangles and simplexes (without fixed coordinate axes) are equivalent, because rectangles and simplexes can be countably approximated to each other
[Fubini-theorem]
Because the piecewise constant value of the rectangular region is product decomposable, and then use the absolutely convergent upper bound to control it
Fubini theorem 2 โฆ (ref-5)
Fubini's theorem can be used to prove that the volume calculation below the graph of a function is the integral of the height function with respect to the volume of the base
Example polar coordinate, 2d, 3d, hyperbolic โฆ
area coarea formula โฆ
[low-dim-integral]
The integral of a form over a -simplex is also invariant, and there's no need to define volume for lower-dimensional simplices.
constant-type form
simplicial map type form := Let be the vertices of the simplex, then
Similar to the -order case, the integral of a simplicial map form on a -chain is defined as
Even if two -simplices are adjacent, their orientations might be discontinuous. This is different from the -order case, where the codimension is zero, so all -simplices have the same orientation.
Two adjacent -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 -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 -direction discontinuity.
[integral-on-submanfold]
let be dimension manifold, form restrict to orientable submanifold tangent space
form in is equivalent to scalar function, but how to control form by integral? try ?
if unnecessary, do not introduce metric to define or now