Exercise 1.7.6 (Finitely additive measure that is not a pre-measure)

Question

Construct a finitely additive measure $\mu_0: {\mathcal B}_0 \rightarrow [0,+\infty]$ that is not a pre-measure.

Elu Thingol · Accepted Answer

Consider the measure $\lambda$ which acts as a infinity test on the power set of natural numbers. In other words,
$$\lambda : 2^\mathbb{N} 	o [0,+\infty], \qquad \lambda(E) = \begin{cases} 0 &	ext{if $E$ is finite}\ 1 &	ext{if $E$ is infinite} \end{cases}.$$
It is easy to verify that we have $\lambda$ is finitely additive, yet the countable subadditivity fails with
$$1 = \lambda\left(\bigcup_{n=1}^{\infty}\{n\}ight) \geq \sum_{n=1}^{\infty} \lambda(\{n\}) = 0.$$

Exercise 1.7.6 (Finitely additive measure that is not a pre-measure)

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Exercise 1.7.6 (Finitely additive measure that is not a pre-measure)

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