Properties of Group Manifolds by Smith P.A.

By Smith P.A.

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A set F ⊂ Ω is said to be Hψ -measurable for some ψ ∈ D if Hψ ∗ (A) = Hψ ∗ (A ∩ F ) + Hψ ∗ (A ∩ F c ) , (91) for all A ⊂ Ω. Here F c denotes the complement Ω\F . The following theorem due to C. Carath´eodory [22] then establishes our measure space. Theorem 54. Let Mψ denote the set of measurable sets in Ω with respect to the outer ∗ ∗ measure Hψ . Then Mψ is a σ-algebra and the restriction Hψ of Hψ to Mψ is a measure. Remark 55. , it also satisfies: Hψ (∪∞ i=1 Fi ) = ∞ Hψ (Fi ), (92) i=1 for any countable family (Fi ) of mutually disjoint measurable subsets of Ω.

F : f is φ-H¨older continuous in F } {φ ∈ F : f is locally φ-H¨older continuous at u ∈ F } (83) {φ ∈ F : f is locally right φ-H¨older continuous at u ∈ F } {φ ∈ F : f is locally left φ-H¨older continuous at u ∈ F } Obviously, the suprema may or may not exist, depending on whether these sets have an upper bound or not, or on whether they are empty or not. As an example, we consider the Weierstrass and the Takagi functions. Lemma 48. The Takagi function is locally ψ-H¨older continuous for ψ(x) = x log with x ∈ 0, e−1 .

1 See chapter 10 of [33] for a survey of applications of fractional calculus and [42, 44]. 1. N. C. Dias and J. N. Prata Local Fractional Derivatives Definition 78. Let f ∈ C 0 (I) and α ∈ ( 0, 1] . The right (σ = +) or left (σ = −) Kolwankar-Gangal local fractional derivative of order α of f at ξ ∈ I is given by: dα f σ 1 (ξ ) := lim Dα [σ (f − f (ξ))] (x). α dx Γ(1 + α) x→ξσ ξ,−σ (116) If the limits exist, we say that f is right or left α-differentiable in the sense of Kolwankar and Gangal (KG) at ξ.

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