General Measure and Integration

The first five sections of this chapter are dedicated to the study of measure theory, measurable functions on a measure space, general integration theory on a measure space, and the interchange of integration and limit via the Lebesgue Dominated Convergence Theorems. I define the concept of topological measure space $$\mathbb {X}$$ X on a topological space $$\mathcal {X} := (X,\mathcal {O})$$ X : = ( X , O ) to be a the space $$((X,\mathcal {O}),\mathcal {B}_{\textrm{B}}{\left( \mathcal {X} \right) },\mu )$$ ( ( X , O ) , B B X , μ ) , where $$\mu $$ μ is a measure that satisfies a certain nice condition. On a topological measure space, all measurable subsets admit a topological measure structure. The Littlewood’s Three Principles are cleanly established and appropriately generalized to Borel measurable spaces. The integration of a measurable function on a measure space is carefully defined. So, $$0 \times \infty = 0$$ 0 × ∞ = 0 is completely banished from the theory. I then define Banach space valued measure space, the integration of Banach space valued function over Banach space valued measure space, and general convergence theorems for these more general integrals. Radon–Nikodym theorem is presented for general Banach space valued measures with respect to real- or complex-valued measure space under the assumption of absolute continuity. $$\textrm{L}_{p}(\mathcal {X},\mathscr {Y})$$ L p ( X , Y ) space is presented with the Riesz Representation Theorem 11.186. This chapter ends with the study of the Riesz Representation Theorem that states that, under certain conditions, we have $$(\mathcal {C}_{}(\mathcal {X},\mathscr {Y}))^* = \mathscr {M}_{ft}(\mathcal {X},\mathscr {Y}^*)$$ ( C ( X , Y ) ) ∗ = M ft ( X , Y ∗ ) (see Theorem 11.201). I solve the puzzle of $$\mathcal {C}_{\textrm{c}}(\mathscr {X},\mathscr {Y})$$ C c ( X , Y ) and prove $$(\mathcal {C}_{\textrm{c}}(\mathscr {X},\mathscr {Y}))^* = \mathscr {M}_{ft}(\mathscr {X},\mathscr {Y}^*)$$ ( C c ( X , Y ) ) ∗ = M ft ( X , Y ∗ ) . This characterization allows me to define the weak $${^*}$$ ∗ convergence in $$\mathscr {M}_{ft}(\mathscr {X},\mathscr {Y}^*)$$ M ft ( X , Y ∗ ) as the notion of convergence in distribution on the space of measures.