Q4.(a) Let > be a o- Algebra on X.Assume that u1, 2 :> → [0, 0) are two measures on X.Is u, + ku, a measure on X? where k is a scalar.Consider the cases: k >0 and k < 0.(b) Let (X, > , µ) be a measure space. For A, Be >.Show that μ (AU B) = μ (A) + μ(B) -μ (Α Β) .

Solution of part (a): Given∑be a σ-Algebra on X. Assume that μ1, μ2 : ∑→[0,∞) are two measures on…

(a) Let > be a o- Algebra on X. Assume that p1, µ2 :)→ [0, 0) are two measures on X. Is µ1 + kµ, a measure on X? where k is a scalar. Consider the cases: k 20 and k < 0. (b) Let (X, > ,4) be a measure space. For A , B e ). Show that u(AUB) = µ(A) + µ(B) – µ(An B). -

(a) Let > be a o- Algebra on X. Assume that p1, µ2 :)→ [0, 0) are two measures on X. Is µ1 + kµ, a measure on X? where k is a scalar. Consider the cases: k 20 and k < 0. (b) Let (X, > ,4) be a measure space. For A , B e ). Show that u(AUB) = µ(A) + µ(B) – µ(An B). -

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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Q4.
(a) Let > be a o- Algebra on X.
Assume that u1, 2 :> → [0, 0) are two measures on X.
Is u, + ku, a measure on X? where k is a scalar.
Consider the cases: k >0 and k < 0.
(b) Let (X, > , µ) be a measure space. For A, Be >.
Show that μ (AU B) = μ (A) + μ(B) -μ (Α Β) .

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