Abstract
The sets {f: f ∈ ℂℝ, f of bounded variation on ℝ} resp. {f: f ∈ℂℝ, f absolutely continuous on ℝ} are denoted BV(ℝ, ℂ) resp. AC(ℝ, ℂ); similar meanings are attached to BV(I, ℂ) resp. AC(I, ℂ). If f ∈ BV(ℝ, ℂ) then T f :x → R is the total variation of f on (-∞, x], i.e., T f (x) = sup{Σni=1 ∣ f (xi+1) - f (xi)∣: n in ℕ, -∞ < x1 < ... <x n +1 = x} and T f (ℝ) = supx T f (x); similar meanings are given to T f (x) and T f (I) if f ∈BV(I, ℂ). The sets [x1, x2), [x2, x3),..., [x n , x] constitute a partition P of [x1, x] and ∣P∣ = supi ∣xi+1 - x1,∣; T fP = Σni=1 ∣ f (xi+1) - f (xi)∣: μ f is the Borel measure such that μ f ([a, b)) = T f ([a, b)).
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© 1982 Springer-Verlag New York Inc.
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Gelbaum, B.R. (1982). Miscellaneous Problems. In: Problems in Analysis. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-7679-2_15
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DOI: https://doi.org/10.1007/978-1-4615-7679-2_15
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4615-7681-5
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