Let X be a non-empty set and F be an arbitrary σ-algebra on X. Define a function μ : F → [0, ∞] by μ(A) = n if A is a finite set with n elements, and μ(A) = ∞ if A is an infinite set. Prove that μ is a measure.

Q: Exercise 4.4.7. Let A be a set. Let Ø : P(A) → P(A) be defined by ø(X) = A − X for is bijective. all…

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Q: The function f : R → R is defined by the rule f(x)=2x+3              a. Prove f(x) is one to one…

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Q: ► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a…

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Q: Let N be a nonempty set and B a o-ring on N. Let µ, v be two measures on B and μ(A) < ν(Α), A € B.…

A: Let Ω be a non- empty set. And B a sigma ring on Ω. Let μ,ν be the two measures on B And ΩA≤νA ,…

Q: Show that a map f : X → Y invertible if and only if f is one-to-one and onto

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Q: Let N C C be a domain (that is, a connected open subset), and let f: N → C be holomorphic. Prove…

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Q: Let X be a Banach space and T = L(X, X) have ||T|| < 1. Define Tº to be the identity map (that is,…

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Q: (a) Assume that f: A onto B and g: B → A. Prove that g = f-¹ iff g of = IA or fog = IB (Theorem…

A: As per the policy we are solving question (a) for you .For the solution of (b) please repost the…

Q: Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is…

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Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

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Q: Let A be a non-empty set and suppose that f : A → Rand g : A → R. Define f + g : A → R by ( ƒ +…

A: we take f(x)=-1   x∈Q∩A1       x∈Qc∩A and g(x)=2   x∈Q∩A-2   x∈Qc∩A(f+g)(x)=f(x)+g(x)=1   x∈Q∩A-1…