By Daniel J. Velleman
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Extra info for American Mathematical Monthly, volume 117, June July 2010
Since A is μ-splitting we have μ [C ∩ (A ∪ B)] = μ [(C ∩ A) ∩ (A ∪ B)] + μ (C ∩ A ) ∩ (A ∪ B) = μ(C ∩ A) + μ(C ∩ A ∩ B). Hence, since B is μ-splitting we conclude that μ(C) = μ(C ∩ A) + μ(C ∩ A ) = μ(C ∩ A) + μ(C ∩ A ∩ B) + μ(C ∩ A ∩ B ) = μ [C ∩ (A ∪ B)] + μ C ∩ (A ∪ B) . It follows that A ∪ B is μ-splitting, so A ∪ B ∈ Z μ . Hence, Z μ is a Boolean subalgebra of P (X ). Moreover, μ | Z μ is a measure because if A, B ∈ Z μ with A ∩ B = ∅, since AμB we have μ(A ∪· B) = μ(A) + μ(B). To prove the last statement, let Ai ∈ Z μ be r mutually disjoint, i = 1, .
Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover, New York, 1975. 12. M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Berlin, 1989. 13. X. Tan, J. S. Baras, and P. S. Krishnaprasad, Control of hysteresis in smart actuators with application to micro-positioning, Systems Control Lett. 54 (2005) 483–492. 09. 013 14. A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. 15. W. Walter, Ordinary Differential Equations, Springer, New York, 1998.
It remains only to show that (RDE2) also holds (with h = 0). Let I ∈ I and v ∈ WG (I ) be such that gr v is a compact subset of G. 1 hold with M = N , q = 1, and g = f . 1, we may infer that I → R N , t → f (t, v(t)) is in L 1 (I ). Therefore, (RDE2) holds. 1. 1 hold. Let I ∈ I and v ∈ W (I ). 1, we conclude q that the function s → g(s, v(s)) is in L loc (I ). It follows now from a routine argument based on Fubini’s theorem (see, for example, [7, 11]) that F(v) ∈ L 1loc (I ). 1 guarantees that the function s → g(s, v(s)) is in L q (I ) and the same routine argument based on Fubini’s theorem yields F(v) ∈ L 1 (I ), showing that (H3) is valid.
American Mathematical Monthly, volume 117, June July 2010 by Daniel J. Velleman