I am reading 1st edition of Folland's Real Analysis at the moment, and I have a question about one of the questions in 1st chapter. Direct quote from folland:
If $\mu$ is a measure on $(X, \mathcal{M})$, define $\mu_0$ on $\mathcal{M}$ by $\mu_0(E)=sup\{\mu(F) : F \subset E,\, \mu(F) < \infty\}$
Then he claims this is a semifinite measure. However, when I think about this, I could find some counterexamples, such as: $X=\mathbb{R}$, $\mathcal{M} = \{\emptyset, \mathbb{R} - \{0\}, \{0\}, \mathbb{R} \}$ and $\mu$ takes $\mathbb{R} - \{0\}$ to $\infty$ and $\{0\}$ to any positive real number. And if I try to define what folland says, I can not get a semifinite measure. Am I doing something wrong? If so, what? If not, what is the bare minimum that should I assume about this first measure to get a semifinite measure?
