Let X be a non-empty set and F be an arbitrary σ-algebra onX. Define a function μ : F → [0, ∞] by μ(A) = n if A is a finite set with n elements, andμ(A) = ∞ if A is an infinite set. Prove that μ is a measure. (That is, show that μ satisfiesDefinition 4.1.15.) ) Let X be a non-empty set and F be an arbitrary o-algebra onX. Define a function u : F [0, o0] by u(A) = n if A is a finite set with n elements, andu(A) = o if A is an infinite set. Prove that u is a measure. (That is, show that u satisfiesDefinition 4.1.15.)

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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.

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.

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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Question

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. (That is, show that μ satisfies
Definition 4.1.15.)

) Let X be a non-empty set and F be an arbitrary o-algebra on
X. Define a function u : F [0, o0] by u(A) = n if A is a finite set with n elements, and
u(A) = o if A is an infinite set. Prove that u is a measure. (That is, show that u satisfies
Definition 4.1.15.)

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