Q14 Let X= R. Define a function : P(R) → [0,0] by #* (A) = { #A (a) Show that is an outer measure on R. #Aif A is finite, if A is infinite. (b) Show that every ACR is -measurable and hence deduce that in fact defines a measure on P(R). (c) Prove that every function f: R→ Ris (P(R), B(R))-measurable, where B(R) denotes the Borelo-algebra on R

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A: The given in this question is a measurable space (X,F) and two F-measurable functions f and g from X…

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Q: 2. Let (X,r) and (Y.A) be two topological spaces. Let B be a base for A. Prove that a function f…

A: Given below the detailed solution

Q: b) Give an example to show that the product fg need not be uniformly continuous.

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A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a<x*<b then…

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A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

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Q: 11. with f(-1)=6, f(1) = 8 and f(6) = 3. Intermediate Value Theorem? Let f be a function that is…

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Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

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Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

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Q: Theorem 18.1 states that "Let f be a continuous real-valued function on a closed interval [a, b].…

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Q: Find the particular solution. y(x) = = e³y = 3(x + 2)y¹, y(0) = - 7 21 x e-3x +49 e-8x48 1 37

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Q: 13 (2). Consider the problem of maximizing the algebraic expression q given by q = V6 - (I+ 2)2 – y?…

A: For the solution of the problem follow the next steps.

Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

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