Measure, Integration & Real Analysis · Problem 2C.11

Measures — Sheldon Axler

Mathematics — Measure Theory · measures

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) is a measure space and C,D,E∈SC, D, E \in \mathcal{S} are such that μ(C∩D)<∞,μ(C∩E)<∞,μ(D∩E)<∞.\mu(C \cap D) < \infty, \quad \mu(C \cap E) < \infty, \quad \mu(D \cap E) < \infty. Find and prove a formula for μ(C∪D∪E)\mu(C \cup D \cup E) in terms of μ(C)\mu(C), μ(D)\mu(D), μ(E)\mu(E), μ(C∩D)\mu(C \cap D), μ(C∩E)\mu(C \cap E), μ(D∩E)\mu(D \cap E), and μ(C∩D∩E)\mu(C \cap D \cap E).

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