Question

In: Advanced Math

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that...

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that ?(?∪?) = ?(?)+?(?)−?(?∩?) where ?,? ∈?.

Expert Solution

Proof:

Let be a algebra in and a measure on .

Given that and are measurable sets.

If either or have infinite measure then the inequality trivially hold.

Assume that and have finite measure.

Write as disjoint union.

By finite additivity and excision property we get

Colby Messinger answered 1 year ago

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