Question 11.7: Let f ∈ L¹(Ω) and define the function ν on M0 = {E ⊆ Ω : E ∈...
Let f ∈ \mathcal{L}^{1}(Ω) and define the function ν on \mathcal{M}_{0} = \left\{E ⊆ Ω : E ∈ \mathcal{M}\right\} by
ν(E) = \int_{E} {f} dm.
Prove that ν is countably additive.
Suppose (E_{n}) is a pairwise disjoint sequence in \mathcal{M}_{0} and E = \cup_{n=1}^{∞} E_{n}. We can then write
f χ_{E} = \sum\limits_{n=1}^{∞}{f χ_{E_{n}}}.
Applying Corollary 11.3.1, we have
\sum\limits_{n=1}^{∞} {\int_{Ω}{\left|f χ_{E_{n}}\right|}dm} = \int_{Ω} \sum\limits_{n=1}^{∞} \left|f\right| χ_{E_{n}} dm
= \int_{Ω} \left|f\right| χ_{E} dm < ∞.
By Corollary 11.3.2,
\int_{Ω} {f χ_{E}} dm = \int_{Ω} \sum\limits_{n=1}^{∞} {f χ_{E_{n}}} dm = \sum\limits_{n=1}^{∞} { \int_{Ω} f χ_{E_{n}}} dm.
This implies ν(E) = \int_{E} {f} dm = \Sigma_{n=1}^{∞} {\int_{E_{n}}} f dm = \Sigma_{n=1}^{∞} {v (E_{n})}.