By Claude Dellacherie

Dellacherie C. Capacites et processus stochastiques (fr)(ISBN 0387056769)(Springer, 1972)

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Fred Almgren created the surplus strategy for proving regularity theorems within the calculus of diversifications. His strategies yielded Holder continuity aside from a small closed singular set. within the sixties and seventies Almgren sophisticated and generalized his tools. among 1974 and 1984 he wrote a 1,700-page facts that used to be his such a lot formidable exposition of his ground-breaking principles.

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Denote the collection of all measurable sets by B. Step 3. The σ-algebra of measurable sets and σ-additivity of the measure. The main part of the proof consists of demonstrating that B is a σ-algebra, and that the function µ deﬁned on it has the properties of a measure. We can then restrict the measure to the smallest σ-algebra containing the original semialgebra. The uniqueness of the measure follows easily from the non-negativity of m and from the fact that the measure is uniquely deﬁned on the algebra A.

Then, almost surely, there exists a ﬁnite limit f (ω) = lim fn (ω), n→∞ the function f is integrable, and Ω f dµ = limn→∞ Ω fn dµ. 29. (Fatou Lemma) If fn is a sequence of non-negative measurable functions, then lim inf fn dµ ≤ lim inf Ω n→∞ n→∞ fn dµ ≤ ∞. Ω Let us discuss products of σ-algebras and measures. Let (Ω1 , F1 , µ1 ) and (Ω2 , F2 , µ2 ) be two measurable spaces with ﬁnite measures. We shall deﬁne the product space with the product measure (Ω, F, µ) as follows. The set Ω is just a set of ordered pairs Ω = Ω1 × Ω2 = {(ω1 , ω2 ), ω1 ∈ Ω1 , ω2 ∈ Ω2 }.

2m 50 3 Lebesgue Integral and Mathematical Expectation Therefore, ∞ ∞ ∞ (Ω\Ωnm0 (m) )) ≤ µ(Ω\Ωδ ) = µ( m=1 µ(Ω\Ωnm0 (m) ) < m=1 δ = δ, 2m m=1 ✷ which completes the proof of the theorem. The following theorem justiﬁes passage to the limit under the sign of the integral. 27. (Lebesgue Dominated Convergence Theorem) If a sequence of measurable functions fn converges to a measurable function f almost everywhere and |fn | ≤ ϕ, where ϕ is integrable on Ω, then the function f is integrable on Ω and lim fn dµ = n→∞ f dµ.