Restricted kinds of densities and associated ideals Pratulananda Das Department of Mathematics, Jadavpur University, West Bengal Pratulananda Das Restricted kinds of densities and associated ideals
THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!! Pratulananda Das Restricted kinds of densities and associated ideals
THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!! • Let ( X , ρ ) be a metric space. Recall the definition of usual convergence of a sequence ( x n ) in X . Pratulananda Das Restricted kinds of densities and associated ideals
THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!! • Let ( X , ρ ) be a metric space. Recall the definition of usual convergence of a sequence ( x n ) in X . Definition 1. A sequence ( x n ) in X is said to converge to x 0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ ( x n , x 0 ) < ε for all n ≥ m . Pratulananda Das Restricted kinds of densities and associated ideals
THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!! • Let ( X , ρ ) be a metric space. Recall the definition of usual convergence of a sequence ( x n ) in X . Definition 1. A sequence ( x n ) in X is said to converge to x 0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ ( x n , x 0 ) < ε for all n ≥ m . • It is well known that a sequence is convergent if and only if one of its tails converges to the same limit. Also for a sequence, the set of all limit points (subsequential limits) forms a closed set. • Consider the following sequences. Pratulananda Das Restricted kinds of densities and associated ideals
THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!! • Let ( X , ρ ) be a metric space. Recall the definition of usual convergence of a sequence ( x n ) in X . Definition 1. A sequence ( x n ) in X is said to converge to x 0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ ( x n , x 0 ) < ε for all n ≥ m . • It is well known that a sequence is convergent if and only if one of its tails converges to the same limit. Also for a sequence, the set of all limit points (subsequential limits) forms a closed set. • Consider the following sequences. Example 1. In a metric space ( X , ρ ) containing at least two points (for example R for simplicity) take two distinct points z and w . Let x n = z when n is even and x n = w when n is odd. Pratulananda Das Restricted kinds of densities and associated ideals
THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!! • Let ( X , ρ ) be a metric space. Recall the definition of usual convergence of a sequence ( x n ) in X . Definition 1. A sequence ( x n ) in X is said to converge to x 0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ ( x n , x 0 ) < ε for all n ≥ m . • It is well known that a sequence is convergent if and only if one of its tails converges to the same limit. Also for a sequence, the set of all limit points (subsequential limits) forms a closed set. • Consider the following sequences. Example 1. In a metric space ( X , ρ ) containing at least two points (for example R for simplicity) take two distinct points z and w . Let x n = z when n is even and x n = w when n is odd. Example 2. Consider another sequence ( y n ) where y n = w if n = k 2 for some k ∈ N and y n = z otherwise. Pratulananda Das Restricted kinds of densities and associated ideals
• In order to understand the difference between the two sequences, we will have to consider the following notion. Pratulananda Das Restricted kinds of densities and associated ideals
• In order to understand the difference between the two sequences, we will have to consider the following notion. Definition 2. By | A | we denote the cardinality of a set A . The lower and the upper natural densities of A ⊂ N are defined by | A ∩ [ 1 , n ] | | A ∩ [ 1 , n ] | d ( A ) = lim inf and d ( A ) = lim sup . n n n →∞ n →∞ If d ( A ) = d ( A ) , we say that the natural density of A exists and it is denoted by d ( A ) . Pratulananda Das Restricted kinds of densities and associated ideals
• In order to understand the difference between the two sequences, we will have to consider the following notion. Definition 2. By | A | we denote the cardinality of a set A . The lower and the upper natural densities of A ⊂ N are defined by | A ∩ [ 1 , n ] | | A ∩ [ 1 , n ] | d ( A ) = lim inf and d ( A ) = lim sup . n n n →∞ n →∞ If d ( A ) = d ( A ) , we say that the natural density of A exists and it is denoted by d ( A ) . Observation: We say that a subset of N is ”small” if it has natural density zero. Evidently any finite set has natural density zero. But now see that the set of odd integers as well as the set of even integers has density 1 2 whereas the set of all squares has evidently density zero (just think that if A = { n : n = k 2 , k ∈ N } then within the the first n 2 positive integers, A has n elements and so the density is equal to lim n n 2 .) Pratulananda Das Restricted kinds of densities and associated ideals
• The following definition was introduced by Fast (1951), Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Pratulananda Das Restricted kinds of densities and associated ideals
• The following definition was introduced by Fast (1951), Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Definition 3. A sequence ( x n ) in ( X , ρ ) is said to be statistically convergent to x 0 ∈ X if for arbitrary ε > 0 the set K( ε ) = { n ∈ N : ρ ( x n , x 0 ) ≥ ε } has natural density zero. Pratulananda Das Restricted kinds of densities and associated ideals
• The following definition was introduced by Fast (1951), Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Definition 3. A sequence ( x n ) in ( X , ρ ) is said to be statistically convergent to x 0 ∈ X if for arbitrary ε > 0 the set K( ε ) = { n ∈ N : ρ ( x n , x 0 ) ≥ ε } has natural density zero. Observation: Evidently it is now clear that the sequence ( y n ) defined above is statistically convergent to z (though is not convergent in the usual sense). Also it is easy to note that statistically convergent sequences need not be bounded as can be seen by observing the sequence ( x n ) of real numbers where x n = n if n = k 2 , k ∈ N and x n = 0, otherwise. Pratulananda Das Restricted kinds of densities and associated ideals
• The following definition was introduced by Fast (1951), Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Definition 3. A sequence ( x n ) in ( X , ρ ) is said to be statistically convergent to x 0 ∈ X if for arbitrary ε > 0 the set K( ε ) = { n ∈ N : ρ ( x n , x 0 ) ≥ ε } has natural density zero. Observation: Evidently it is now clear that the sequence ( y n ) defined above is statistically convergent to z (though is not convergent in the usual sense). Also it is easy to note that statistically convergent sequences need not be bounded as can be seen by observing the sequence ( x n ) of real numbers where x n = n if n = k 2 , k ∈ N and x n = 0, otherwise. • Statistical convergence again resurfaced through the major works by Salat (1980), Fridy (1885) and Connor (1989) and from then on the related investigations has become the most active research areas in Summability theory Pratulananda Das Restricted kinds of densities and associated ideals
• We start by recalling the basic notions of ideals and filters. Pratulananda Das Restricted kinds of densities and associated ideals
• We start by recalling the basic notions of ideals and filters. Definition 4. A family I ⊂ 2 Y of subsets of a non-empty set Y is said to be an ideal in Y if ( i ) A , B ∈ I implies A ∪ B ∈ I , ( ii ) A ∈ I , B ⊂ A imply B ∈ I . Pratulananda Das Restricted kinds of densities and associated ideals
• We start by recalling the basic notions of ideals and filters. Definition 4. A family I ⊂ 2 Y of subsets of a non-empty set Y is said to be an ideal in Y if ( i ) A , B ∈ I implies A ∪ B ∈ I , ( ii ) A ∈ I , B ⊂ A imply B ∈ I . Observation: Further an admissible ideal I of Y satisfies { x } ∈ I for each x ∈ Y . Such ideals are also called free ideals. If I is a proper non-trivial ideal in Y (i.e. ∈ I , I � = { φ } ) , then the family of sets F ( I ) = { M ⊂ Y : Y / there exists A ∈ I : M = Y \ A } is a filter in Y . It is called the filter associated with the ideal I . Pratulananda Das Restricted kinds of densities and associated ideals
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