Completeness in topological vector spaces and filters on $\mathbb N$
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Ideal on an arbitrary non-empty set $\Omega$ it's a non-empty family of subset $\mathfrak{I}$ of the set $\Omega$ which satisfies the following axioms: $\Omega \notin \mathfrak{I}$, if $A, B \in \mathfrak{I}$, then $A \cup B \in \mathfrak{I}$, if $A \in \mathfrak{I}$ and $D \subset A$, then $D \in \mathfrak{I}$. The ideal theory is a very popular branch of modern mathematical research. In our paper we study some classes of ideals on the set of all positive integers $\mathbb{N}$, namely the ideal of statistical convergence $\mathfrak{I}_s$ and the ideal $\mathfrak{I}_f$ generated by a modular function $f$. Statistical ideal it's a family of subsets of $\mathbb{N}$ whose natural density is equal to 0, i.e. $A \in \mathfrak{I}_s$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{\#\{k \leq n: k \in A\}}{n} = 0$. A function $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ is called a modular function, if $f(x) = 0$ only if $x = 0$, $f(x + y) \leq f(x) + f(y)$ for all $x, y \in\mathbb{R}^+$, $f(x) \le f(y)$ whenever $x \le y$, $f$ is continuous from the right 0, and finally $\lim\limits_{n \rightarrow \infty} f(n) = \infty$. Ideal, generated by the modular function $f$ it's a family of subsets of $\mathbb{N}$ with zero $f$-density, in other words, $A \in \mathfrak{I}_f$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{f(\#\{k \leq n: k \in A\})}{f(n)} = 0$. It is known that for an arbitrary modular function $f$ the following is true: $\mathfrak{I}_f \subset \mathfrak{I}_s$. In our research we give the complete description of those modular functions $f$ for which $\mathfrak{I}_f = \mathfrak{I}_s$. Then we analyse obtained result, give some partial cases of it and prove one simple sufficient condition for the equality $\mathfrak{I}_f = \mathfrak{I}_s$. The last section of this article is devoted to examples of some modulus functions $f, g$ for which $\mathfrak{I}_f = \mathfrak{I}_s$ and $\mathfrak{I}_g \neq \mathfrak{I}_s$. Namely, if $f(x) = x^p$ where $p \in (0, 1]$ we have $\mathfrak{I}_f = \mathfrak{I}_s$; for $g(x) = \log(1 + x)$, we obtain $\mathfrak{I}_g \neq \mathfrak{I}_s$. Then we consider more complicated function $f$ which is given recursively to demonstrate that the conditions of the main theorem of our paper can't be reduced to the sufficient condition mentioned above.
Ideal on an arbitrary non-empty set $\Omega$ it's a non-empty family of subset $\mathfrak{I}$ of the set $\Omega$ which satisfies the following axioms: $\Omega \notin \mathfrak{I}$, if $A, B \in \mathfrak{I}$, then $A \cup B \in \mathfrak{I}$, if $A \in \mathfrak{I}$ and $D \subset A$, then $D \in \mathfrak{I}$. The ideal theory is a very popular branch of modern mathematical research. In our paper we study some classes of ideals on the set of all positive integers $\mathbb{N}$, namely the ideal of statistical convergence $\mathfrak{I}_s$ and the ideal $\mathfrak{I}_f$ generated by a modular function $f$. Statistical ideal it's a family of subsets of $\mathbb{N}$ whose natural density is equal to 0, i.e. $A \in \mathfrak{I}_s$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{\#\{k \leq n: k \in A\}}{n} = 0$. A function $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ is called a modular function, if $f(x) = 0$ only if $x = 0$, $f(x + y) \leq f(x) + f(y)$ for all $x, y \in\mathbb{R}^+$, $f(x) \le f(y)$ whenever $x \le y$, $f$ is continuous from the right 0, and finally $\lim\limits_{n \rightarrow \infty} f(n) = \infty$. Ideal, generated by the modular function $f$ it's a family of subsets of $\mathbb{N}$ with zero $f$-density, in other words, $A \in \mathfrak{I}_f$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{f(\#\{k \leq n: k \in A\})}{f(n)} = 0$. It is known that for an arbitrary modular function $f$ the following is true: $\mathfrak{I}_f \subset \mathfrak{I}_s$. In our research we give the complete description of those modular functions $f$ for which $\mathfrak{I}_f = \mathfrak{I}_s$. Then we analyse obtained result, give some partial cases of it and prove one simple sufficient condition for the equality $\mathfrak{I}_f = \mathfrak{I}_s$. The last section of this article is devoted to examples of some modulus functions $f, g$ for which $\mathfrak{I}_f = \mathfrak{I}_s$ and $\mathfrak{I}_g \neq \mathfrak{I}_s$. Namely, if $f(x) = x^p$ where $p \in (0, 1]$ we have $\mathfrak{I}_f = \mathfrak{I}_s$; for $g(x) = \log(1 + x)$, we obtain $\mathfrak{I}_g \neq \mathfrak{I}_s$. Then we consider more complicated function $f$ which is given recursively to demonstrate that the conditions of the main theorem of our paper can't be reduced to the sufficient condition mentioned above.
This article is devoted to the study of one generalization of the Riemann integral. Namely, in the paper, it was observed that the classical definition of the Riemann integral over a finite segment as a limit of integral sums, when the diameter of the division of the segment tends to zero, can be replaced by a limit of integral sums over a filter of sets, which can be described in a certain "good way". This idea was continued, and in the work we propose a new concept - the integral of a function over a filter on the set of all tagged partitions of a segment. Using of filters is a very good method in questions related to convergence or some of its analogues in general topological vector spaces. Namely, if the space is non-metrizable, then the concept of convergence is introduced precisely with the help of filters. Also, using filters, you can formulate the concept of completeness and its analogues. The completeness of spaces is one of the central concepts of the theory of topological vector spaces, since Banach spaces are complete. That is, using a generalization of the completeness of spaces constructed using filters, we can explore various generalizations of Banach spaces. We study standard issues related to integration. For example, does the integrability of the filter function imply its boundedness? The answer to this question is affirmative. Namely: the concept of filter boundedness of a function is introduced, and it is shown that if a function is integrable over filter, then its integral sums are bounded over the filter, and this function itself is bounded in the classical sense. Next, we showed that the filter integral satisfies the linearity property, namely, the integral over filter of the sum of two functions is the sum of the filter integrals of these functions. In addition, we can to subtract the constant factor from the sign of the integral over filter. We introduce the concept of an exactly tagged filter, and with the help of such filters we study the filter integrability of unbounded functions on a segment. We give an example of a specific unbounded function and a specific filter under which this function is integrable. Next, we prove a theorem that describes unbounded filter-integrable functions on a segment. The last section of the article is devoted to the integration of functions relative to the filter on a subsegment of this segment.
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