msleziak.com

Let $A\subset\mathbb R$ be measurable, $\lambda$ stands for Lebesgue measure on $\mathbb R$. We say that $0$ is Lebesgue right density point of $A$ if
$$d^+(A,0) = \lim_{h\to0^+} \frac{\lambda(A\cap[0,h])}h=1.$$
Note that
$$
\frac{\lambda(A\cap[0,\frac1{n+1}])}{\frac1{n+1}}\cdot\frac{n}{n+1} \le \frac{\lambda(A\cap[0,h])}h \le \frac{\lambda(A\cap[0,\frac1n])}{\frac1n}\cdot\frac{n+1}n.
$$
Therefore $d^+(A,0)=1$ if and only if
$$\lim\limits_{n\to\infty} n\lambda\left(A\cap[0,1/n]\right)=1.$$
Let $a_n=\lambda(A\cap[1/(n+1),1/n])$. Then
$$\lambda(A\cap[0,1/m]) = \sum_{k=n}^\infty \lambda(A\cap[1/(k+1),1/k])=\sum_{k=n}^\infty a_k.$$
The intuition is the following. If $d^+(A,0)=1$, then $\lambda(A\cap[1/(n+1),1/n])$ should be close to the length $1/(n(n+1))$ of $[1/(n+1),1/n]$ if $n$ tends to $\infty$, and thus $a_n/(1/n-1/(n+1))=n(n+1)a_n$ should tend to $1$.

In fact the following holds.
Theorem 1. $d^+(A,0)=1$ if and only if $n(n+1)a_n$ tends statistically to $1$.

Theorem 1 justifies the following definition. Let $\mathcal I$ be an ideal of subsets of $\mathbb N$. We say that $x$ is an $\mathcal I$-right-density point of a measurable set $A$, in symbols $d_+^{\mathcal I}(A,x)=1$, if $n(n+1)\lambda(A\cap[x+\frac1{n+1},x+\frac1n])$ tends to $1$ with respect to $\mathcal I$. Theorem 1 says that $d_+^{{\mathcal I}_d}(A,x)=1$ $\Longleftrightarrow$ $d_+(A,x)$ where $\mathcal I_d$ stands for the density zero ideal. By $\mathrm{Fin}$ we denote the ideal of finite subsets of $\mathbb N$. Note that if $\mathcal I \subset \mathcal J$, then $d_+^{\mathcal I}(A,x)=1$ implies $d_+^{\mathcal J}(A,x)=1$. The following is the strengthening of Lebesgue's one-dimensional density theorem. We mimic the proof of Lebesgue's density theorem presented by Faure in [1].

Theorem 2. Let $A\subseteq\mathbb R$ be measurable. Then $\lambda(\{x\in A \colon d_+^{\mathrm{Fin}}(A,x)\ne1\})=0$.

We will discuss the result of Wilczynski [2] where the other connection between Lebesgue density points and the statistical convergence is given there.

[1] C. A. Faure, A short proof of Lebesgue's density theorem. Amer. Math. Monthly 109 (2002), no. 2, 194-196. JSTOR, Google
[2] W. Wilczynski. Statistical density points. Folia Math. 11 (2014), no. 1, 59-62. http://fm.math.uni.lodz.pl/artykuly/11/wilczynski.pdf

Let $\mathcal I$ be an ideal on $\mathbb N$ with the Baire property and $\sum x_n$ be an absolutely convergent series with real terms. We show that the set of $\mathcal I$ -convergent subseries and $\mathcal I$-convergent rearrangements are of the first category in the respective Polish spaces. Further, we use this fact in analysis of series of real-valued functions with the Baire property assuming that $\mathcal I$ is analytic or coanalytic.

We investigate the Baire category of $\mathcal I$-convergent subsequences and rearrangement of a divergent sequence $s=(s_n)$ of reals if $\mathcal I$ is an ideal on $\mathbb N$ having the Baire property. We also discuss the measure of the set of $\mathcal I$-convergent subsequences for some classes of ideals on $\mathbb N$. Our result generalizes theorems due to H. Miller and C. Orhan (cf. [8], 2001).

Moreover, for a sequence of functions $(f_n)$ (from a Polish space $X$ into a complete metric space $Z$) with the Baire property, which is divergent on a comeager set, we investigate the Baire category of $\mathcal I$-convergent subsequences and rearrangements of $(f_n)$ if $\mathcal I$ is an analytic or coanalytic ideal on $\mathbb N$.

Our results generalize a theorem of Kallman (cf. [5], 1999). A similar theorem for subsequences is obtained if $(X,\mu)$ is a $\sigma$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\mathcal I$-divergent $\mu$-alsmost everywhere. Then the set of subsequences of $(f_n)$, $\mathcal I$-divergent $\mu$-almost everywhere, is of full product measure in $\{0,1\}^{\mathbb N}$ (here we ssume in addition that $\mathcal I$ has the property (G)).

[1] M. Balcerzak, Sz. Glab, J. Swaczyna, Ideal invariant injections, submitted.
[2] K. Dems, On I-Cauchy sequences, Real Anal. Exchange 30 (2004/2005), 123–128.
[3] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
[4] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313.
[5] R.R. Kallman, Subsequences and category, Internat. J. Math. 22 (1999), 709–712.
[6] A.S. Kechris, Classical Descriptive Set Theory, Springer, New York 1998.
[7] P. Kostyrko, T. Šalát, W. Wilczynski, I-convergence, Real Anal. Exchange 26 (2000-2001), 669–685.
[8] H.I. Miller, C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar. 93 (2001), 135-151.
[9] S. Solecki: Analytic ideals and their applications, Ann. Pure. Appl. Logic 99 (1999), 51-72
[10] S.M. Srivastava, A course of Borel sets, Springer, New York 1998.