A new theorem in the theory of first return representations of Baire class one functions of a real variable is presented which has as immediate consequences several known characterizations of standard subclasses of the Baire one functions. Further, this theorem yields new insights into how finely Baire one functions can be recovered and yields a characterization of another subclass of Baire one functions.
Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider...
“The kernel functor” from the category of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In , monics are one-to-one, but not necessarily so in LFrm. An embedding for which is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The -objects every embedding of which is KI are characterized;...
For non-empty topological spaces X and Y and arbitrary families ⊆ and we put =f ∈ : (∀ A ∈ )(f[A] ∈ . We examine which classes of functions ⊆ can be represented as . We are mainly interested in the case when is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class (X,ℝ) is not equal to for any ⊆ and ⊆ (ℝ). Thus, (X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of...
Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions...
We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.
ℒ denotes the Lebesgue measurable subsets of ℝ and denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...