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Page 71
... f I 2. 7.5. Let f(oc) I 0 for iv 5 0 and f(iv) I 1 for a; > 0. Define F(zv) I fowf, at 6 R. Find a formula for Where is F continuous? Where is F differentiable? Where is F'(a;) I 7.6. Prove each of the following statements about a ...
... f I 2. 7.5. Let f(oc) I 0 for iv 5 0 and f(iv) I 1 for a; > 0. Define F(zv) I fowf, at 6 R. Find a formula for Where is F continuous? Where is F differentiable? Where is F'(a;) I 7.6. Prove each of the following statements about a ...
Page 85
... let G be the equivalence relation induced by f, and let H be an equivalence relation in A which is coarser than G. Define the image ofH as follows: ̄f(H) = {(f(x), f(y)) : (x,y) ∈ H} Prove that ̄f (H) is an equivalence relation in B. 2 ...
... let G be the equivalence relation induced by f, and let H be an equivalence relation in A which is coarser than G. Define the image ofH as follows: ̄f(H) = {(f(x), f(y)) : (x,y) ∈ H} Prove that ̄f (H) is an equivalence relation in B. 2 ...
Page 143
Theory and Practice John Srdjan Petrovic. Problems 6.2.1. Let I 2x2 — x, P I {0, %, 1, Find L(f, P) and U(f, P). 6.2.2. In Lemma 6.2.3, prove that L(f, P2) 2 L(f, P1). 6.2.3. Suppose that f is integrable on [a, b] and that I: dx > 0 ...
Theory and Practice John Srdjan Petrovic. Problems 6.2.1. Let I 2x2 — x, P I {0, %, 1, Find L(f, P) and U(f, P). 6.2.2. In Lemma 6.2.3, prove that L(f, P2) 2 L(f, P1). 6.2.3. Suppose that f is integrable on [a, b] and that I: dx > 0 ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ