## Characterization of sigma-finite measure spaces

Let be a measure space and be a positive measure on it; prove that contains a strictly positive function if and only if is -finite with respect to .

*Ambrosio et al., Functions of Bounded Variation and Free Discontinuity Problems, Ex 1.5*

**Proof:** Suppose contains a strictly positive function . Then it is easy to see that forms a partition of into sets of finite measure.

Conversely, suppose that is -finite and that are the finite measure sets which partition . Define

and note that is strictly positive, it is well defined since the series is always convergent and moreover, its integral is finite. This may seem as an abstract guess, but, in fact, is very natural.

Here are a few steps I took into finding this example:

- (i) Since the only hypothesis we have is that is -finite, we must use the finite measure sets which decompose to construct a function, and how can we do that if not by using their characteristic functions?
- (ii) Since the desired function must be strictly positive we could think of our function as a series of , but we quickly see that it may happen that both and may not have finite integrals, since the measures of may be finite, but not small enough.
- (iii) To fix this divide every term of the sum by a number bigger than and voilà.

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Categories: Measure Theory, Real Analysis
measure

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