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Extra info for An elementary treatise on calculus of variations (1850)
Finally to make y n a subnet of xi , let gnj (α) = fij (α) if α ∈ Ai , i ≥ j, and specify gnj (α) ∈ Aj arbitrarily if α ∈ Ai for i < j. ) Since y n is a subnet of xi , the net yαn (i) converges for all indices i < n. Let (xn , An ) be a subnet of (y n , Bn ) that converges in position n. By induction we obtain, for the ordinal N + 1, a subnet yα = yαN +1 that converges in all coordinates. This means yα converges in X. Axiom of Choice. The use of the Axiom of Choice in the preceding proof cannot be dispensed with, in the strong sense that Tychonoff’s theorem implies the Axiom of Choice.
Since µ is an extreme point, it follows that µ = σ = τ . e. on E, a contradiction. It follows that µ is a delta-mass supported on a single point. But the µ is not in A⊥ , since it pairs nontrivially with the constant function in A. Haar measure. As a further application of convexity, we now develop the Kakutani fixed-point theorem and use it to prove the existence of Haar measure on a compact group. Our treatment follows Rudin, Functional Analysis, Chapter 5. 11 (Milman) Let K ⊂ X be a compact subset of a Banach space and suppose H = hull(K) is compact.
Let J1 , J2 , . . be a list of the intervals with rational endpoints in [0, 1]. Inductively define B11 = B(J1 ) and B1i+1 = B(Jk ) for the first k such that Jk is disjoint from B11 , . . , B1i . Then every Jk meets some B1i so B1i is dense. Similarly, we can find disjoint second moves that are dense in B1i for each i. Putting all these together, we obtain moves B2i , each contained in some B1i , that are also dense in [0, 1]. Continuing in this way, we obtain a sequence Bki such that Uk = i Bki is dense in [0, 1].