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
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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. (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.)
Transcribed Image Text:) 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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