Characterisation of finite measures

Calculus Level 4

Let μ \mu be a measure on ( R , B ( R ) ) (\mathbb{R},\mathcal{B}(\mathbb{R})) (Borel σ \sigma -algebra) such that ∀ X ∈ B ( R ) , μ ( X ) < ∞ or μ ( R ∖ X ) < ∞ \forall X\in\mathcal{B}(\mathbb{R}), \mu(X)<\infty\text{ or }\mu(\mathbb{R}\setminus X)<\infty Is the measure μ \mu finite?

Yes unconditionally Yes iff μ \mu is σ \sigma -finite No even if μ \mu is σ \sigma -finite

Maxence Seymat by Maxence Seymat , 22, France

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1 solution

Maybe I'm mistaken but I think this is trivial. Any sigma algebra over a set S S must contain that set S S by definition. Therefore, the borel sigma algebra contains R \mathbb{R} , which implies μ ( R ) < ∞ \mu (\mathbb{R}) < \infty . This implies μ \mu is finite by monotonicty of measures.

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Why μ ( R ) < ∞ \mu(\mathbb{R})<\infty ? This isn't part of the hypotheses and not all measures over Borel σ \sigma -algebras are finite. For any measure μ \mu , μ ( ∅ ) = 0 \mu(\emptyset)=0 so you don't have directly information about μ ( R ) \mu(\mathbb{R}) . Try to think of an infinite Dirac measure.

Maxence Seymat - 1 year, 4 months ago

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