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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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.)
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