- #1
Anixx
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Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants.
I will provide introduction in very condensed form to get quicker to the essense.
Conservative part
First of all, let us introduce a concept of numerocity of a set, starting from the subsets of integers.
##N(S)=\sum_{k=-\infty}^\infty p_s(k)=p_s(0)+\sum_{k=1}^\infty p_s(k)+\sum_{k=1}^\infty p_s(-k)##
here, ##p_s## is the indicator function of ##S##. As we can see, in general, numerocity is a divergent series, characterized by the rate of growth and regularized value.
The concept of numerocity is, unlike cardinality, additive. Numerocity of union of two non-intersecting sets is the sum of the numerocities of each one (Euclid's principle as opposed to Cantor-Hume principle implemented in cardinality).
Now, what if we want to generalize a numerocity to other countable sets, which are not necessarily subsets of integers? For instance, we need to count the numerocity of of the roots of a given function. It turns out, that the following expression of distributions often works: ##\delta(f(x))|f'(x)|##. The following integral ##\int_{(a,b)}\delta(f(x))|f'(x)|dx## gives the number of roots of the function ##f(x)## on the interval ##(a,b)##, so we take this integral (which also can be divergent if the numerocity of the roots is infinite) as the numerocity of the roots.
We will denote ##\overline{\delta}(x)=\delta(f(x))|f'(x)|## as the "squarable delta function", and generalize it to other cases via the definition that ##\int_{S}\overline{\delta}(f(x))dx## gives the number of roots of ##f(x)## on ##S##.
Unlike the conventional Delta distribution, squarable delta function satisfies the property ##\overline{\delta}(a x)=\overline{\delta}(x)##, so it allows piece-wise definition.
Via Fourier and Laplace transforms we can see that ##\frac 1\pi\int_0^\infty dx=\overline{\delta}(0)##, so we can define it piecewise
##\overline{\delta}(x)={\begin{cases}\frac 1\pi\int_0^\infty dx,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0.\end{cases}}##
or, more generally,
##\overline{\delta}(x)^p={\begin{cases}\frac p{\pi^p}\int_0^\infty x^{p-1}dx,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0.\end{cases}}##
Here we introduce some more notations. First, we introduce the constant ##\omega=\pi \overline{\delta}(0)=\int_0^\infty dx##. We can see this constant as the germ on infinity of the function ##f(x)=x##. Since the constant is equal to ##\frac12 \sum_{k=-\infty}^\infty1##, it is half of the numerocity of integers, equal to the numerocity of even or odd numbers: ##\omega=1/2 N(\mathbb Z)##. This ##\omega## is the most simple infinite quantity, which can be encountered in the theory of formal power series or Hardy fields.
Using this constant, we can express numerocities in integral form: ##N(S)=\frac1\pi\int_S p_s(x)\omega dx##.
Second notation to introduce is ##u(x)##, a unit impulse function, or discrete delta function: ##u(x)=0^{|x|}##, it is equal to ##1## at zero and otherwise, ##0##. So, ##\overline{\delta}(x)=u(x)\omega##.
Third note on notation is that we will understand ##\int_a^b=\frac12 \int_{(a,b)}+\frac12\int_{[a,b]}##, in other words, the limits of an integral are half-included in the set over which we integrate.
Using these new notation we can write down the general rule for ##\omega##:
##\int_{-1}^1 u(x)f(\omega ) \, dx=f'(\omega )## for any analytic ##f##. For functions, satisfying the requirements of Hardy filds (that is, which have germs at infinity), it is also equal to ##f'(0)+\int_0^\infty f''(x)dx##.
An interesting consequence is that ##\int_{-1}^1 u(x)e^\omega dx=e^\omega##.
Novel part
A question: what is the numerocity of an uncountable set, say an interval ##[0,1)##?
We can express this numerocity using an integral:
##N([0,1))=\int_0^1 \omega dx##
But this quantity cannot be represented as analytic function or power series of ##\omega## or germ of a real function at infinity because otherwise it could be equal to a numerocity of a countable set. In this sense, it is different from cardinal arithmetic, where ##2^{\aleph_0}=\aleph_1##.
In this light, we introduce another constant, ##\alpha##, as a numerocity of interval ##[0,\pi)##:
##\alpha=N([0,\pi))=\pi N([0,1))=\int_0^\pi \overline{\delta}(0)dx=\int_0^1 \omega dx##
For some reason, chosing the interval ##[0,\pi)## instead of ##[0,1)## simplifies further expressions.
The regularized value (finite part) of numerocity of a set is equal to its Euler's characteristic, thus, the regularized value of ##\alpha## is ##0##.
We can now express the numerocity of reals: ##N(\mathbb{R})=N(\mathbb Z)\cdot N([0,1))=\frac{2\omega\alpha}\pi=\frac1\pi\int_{-\infty}^\infty \omega dx##.
Like ##\omega##, ##\alpha## does not behave under integrals like a finite constant. We need to derive rules of ealing with this constant under integrals.
From the rule that numerocity of Cartesian product of two sets is the product of numerocities, we get:
##\int_0^1 \omega \alpha ^n \, dx=\alpha ^{n+1}##
More generaly,
##\int_0^1 \omega f(\alpha) \, dx=\alpha f(\alpha) \tag1##
also
##\int_0^1 f(\omega ) \, dx=\alpha f'(\omega )+f(0)\tag2##
and more generally,
##\int_0^1 f(\omega ) g(x) \, dx=\left(\alpha f'(\omega )+f(0)\right) \int_0^1 g(x) \, dx \tag3##
Some examples part
Introducing yet another constant ##\lambda=-\ln\omega-\gamma=-\int_0^1\frac1x dx## (which due to the last expression we also can call "logarithm of zero"), from the rule ##(3)## we can derive interesting equalities:
##\int_0^1 \frac{\omega }{x} \, dx=-\lambda\alpha##
##\int_0^{\infty } \ln \omega \, dx=\alpha + \lambda \omega##
##\int_0^{\infty } \lambda \, dx=\omega \ln \omega -\alpha##
Since ##\lambda\omega=-\int_0^{\infty } | \tan x| \, dx##, we can obtain an interesting expression for ##\alpha##:
##\alpha=\int_0^{\infty } | \tan x| \, dx+\int_0^{\infty } \ln \omega \, dx##
(here the first term is countably-infinite, less than ##\omega^2##, for instance, and the second term is uncountably-infinite, thus, much greater)
-----------------
The question is: how can we expand these rules of integration involving ##\alpha## to cover all possible cases? Particularly,
* How to reconcile formula ##(1)## and formula ##(3)## so to cover expressions of the form ##\int_0^1 f_1(\alpha)f_2(\omega)g(x)dx##?
* What is the value of ##\int_{-1}^1 u(x)\alpha dx##?
* Will integral ##\int_0^1 \alpha dx## produce a numerocity of a set of cardinality higher than continuum, that cannot be expressed as a power series of ##\alpha##? Is this hierarchy infinite?
I will provide introduction in very condensed form to get quicker to the essense.
Conservative part
First of all, let us introduce a concept of numerocity of a set, starting from the subsets of integers.
##N(S)=\sum_{k=-\infty}^\infty p_s(k)=p_s(0)+\sum_{k=1}^\infty p_s(k)+\sum_{k=1}^\infty p_s(-k)##
here, ##p_s## is the indicator function of ##S##. As we can see, in general, numerocity is a divergent series, characterized by the rate of growth and regularized value.
The concept of numerocity is, unlike cardinality, additive. Numerocity of union of two non-intersecting sets is the sum of the numerocities of each one (Euclid's principle as opposed to Cantor-Hume principle implemented in cardinality).
Now, what if we want to generalize a numerocity to other countable sets, which are not necessarily subsets of integers? For instance, we need to count the numerocity of of the roots of a given function. It turns out, that the following expression of distributions often works: ##\delta(f(x))|f'(x)|##. The following integral ##\int_{(a,b)}\delta(f(x))|f'(x)|dx## gives the number of roots of the function ##f(x)## on the interval ##(a,b)##, so we take this integral (which also can be divergent if the numerocity of the roots is infinite) as the numerocity of the roots.
We will denote ##\overline{\delta}(x)=\delta(f(x))|f'(x)|## as the "squarable delta function", and generalize it to other cases via the definition that ##\int_{S}\overline{\delta}(f(x))dx## gives the number of roots of ##f(x)## on ##S##.
Unlike the conventional Delta distribution, squarable delta function satisfies the property ##\overline{\delta}(a x)=\overline{\delta}(x)##, so it allows piece-wise definition.
Via Fourier and Laplace transforms we can see that ##\frac 1\pi\int_0^\infty dx=\overline{\delta}(0)##, so we can define it piecewise
##\overline{\delta}(x)={\begin{cases}\frac 1\pi\int_0^\infty dx,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0.\end{cases}}##
or, more generally,
##\overline{\delta}(x)^p={\begin{cases}\frac p{\pi^p}\int_0^\infty x^{p-1}dx,&{\text{if }}x=0\\ 0,&{\text{if }}x\ne 0.\end{cases}}##
Here we introduce some more notations. First, we introduce the constant ##\omega=\pi \overline{\delta}(0)=\int_0^\infty dx##. We can see this constant as the germ on infinity of the function ##f(x)=x##. Since the constant is equal to ##\frac12 \sum_{k=-\infty}^\infty1##, it is half of the numerocity of integers, equal to the numerocity of even or odd numbers: ##\omega=1/2 N(\mathbb Z)##. This ##\omega## is the most simple infinite quantity, which can be encountered in the theory of formal power series or Hardy fields.
Using this constant, we can express numerocities in integral form: ##N(S)=\frac1\pi\int_S p_s(x)\omega dx##.
Second notation to introduce is ##u(x)##, a unit impulse function, or discrete delta function: ##u(x)=0^{|x|}##, it is equal to ##1## at zero and otherwise, ##0##. So, ##\overline{\delta}(x)=u(x)\omega##.
Third note on notation is that we will understand ##\int_a^b=\frac12 \int_{(a,b)}+\frac12\int_{[a,b]}##, in other words, the limits of an integral are half-included in the set over which we integrate.
Using these new notation we can write down the general rule for ##\omega##:
##\int_{-1}^1 u(x)f(\omega ) \, dx=f'(\omega )## for any analytic ##f##. For functions, satisfying the requirements of Hardy filds (that is, which have germs at infinity), it is also equal to ##f'(0)+\int_0^\infty f''(x)dx##.
An interesting consequence is that ##\int_{-1}^1 u(x)e^\omega dx=e^\omega##.
Novel part
A question: what is the numerocity of an uncountable set, say an interval ##[0,1)##?
We can express this numerocity using an integral:
##N([0,1))=\int_0^1 \omega dx##
But this quantity cannot be represented as analytic function or power series of ##\omega## or germ of a real function at infinity because otherwise it could be equal to a numerocity of a countable set. In this sense, it is different from cardinal arithmetic, where ##2^{\aleph_0}=\aleph_1##.
In this light, we introduce another constant, ##\alpha##, as a numerocity of interval ##[0,\pi)##:
##\alpha=N([0,\pi))=\pi N([0,1))=\int_0^\pi \overline{\delta}(0)dx=\int_0^1 \omega dx##
For some reason, chosing the interval ##[0,\pi)## instead of ##[0,1)## simplifies further expressions.
The regularized value (finite part) of numerocity of a set is equal to its Euler's characteristic, thus, the regularized value of ##\alpha## is ##0##.
We can now express the numerocity of reals: ##N(\mathbb{R})=N(\mathbb Z)\cdot N([0,1))=\frac{2\omega\alpha}\pi=\frac1\pi\int_{-\infty}^\infty \omega dx##.
Like ##\omega##, ##\alpha## does not behave under integrals like a finite constant. We need to derive rules of ealing with this constant under integrals.
From the rule that numerocity of Cartesian product of two sets is the product of numerocities, we get:
##\int_0^1 \omega \alpha ^n \, dx=\alpha ^{n+1}##
More generaly,
##\int_0^1 \omega f(\alpha) \, dx=\alpha f(\alpha) \tag1##
also
##\int_0^1 f(\omega ) \, dx=\alpha f'(\omega )+f(0)\tag2##
and more generally,
##\int_0^1 f(\omega ) g(x) \, dx=\left(\alpha f'(\omega )+f(0)\right) \int_0^1 g(x) \, dx \tag3##
Some examples part
Introducing yet another constant ##\lambda=-\ln\omega-\gamma=-\int_0^1\frac1x dx## (which due to the last expression we also can call "logarithm of zero"), from the rule ##(3)## we can derive interesting equalities:
##\int_0^1 \frac{\omega }{x} \, dx=-\lambda\alpha##
##\int_0^{\infty } \ln \omega \, dx=\alpha + \lambda \omega##
##\int_0^{\infty } \lambda \, dx=\omega \ln \omega -\alpha##
Since ##\lambda\omega=-\int_0^{\infty } | \tan x| \, dx##, we can obtain an interesting expression for ##\alpha##:
##\alpha=\int_0^{\infty } | \tan x| \, dx+\int_0^{\infty } \ln \omega \, dx##
(here the first term is countably-infinite, less than ##\omega^2##, for instance, and the second term is uncountably-infinite, thus, much greater)
-----------------
The question is: how can we expand these rules of integration involving ##\alpha## to cover all possible cases? Particularly,
* How to reconcile formula ##(1)## and formula ##(3)## so to cover expressions of the form ##\int_0^1 f_1(\alpha)f_2(\omega)g(x)dx##?
* What is the value of ##\int_{-1}^1 u(x)\alpha dx##?
* Will integral ##\int_0^1 \alpha dx## produce a numerocity of a set of cardinality higher than continuum, that cannot be expressed as a power series of ##\alpha##? Is this hierarchy infinite?