1 Measure Space
1.1 Constructing Measure Spaces
The primary aim of measure theory is to define the concept of measurability and ultimately assign a measure to measurable sets. That is, for general \(X\) we wish to construct a function \(\mu:2^{X}\to[0,\infty]\) that assigns each \(E\subset X\) a number \(\mu(E)\in[0,\infty]\), the measure of \(E\) where \(\mu\) satisfies:
- For disjoint \(E_{1}, E_{2}, \dots \in X\) we have \(\mu\left( \bigcup_{j=1}^\infty E_{j} \right)=\sum_{j=1}^\infty \mu(E_{j})\) (countably additive);
- For interval \([a,b]\) we have \(\mu([a,b])=b-a\); and
- For set \(E\subset \mathbb{R}\) and constant \(c\in \mathbb{R}\) we have \(\mu(E+c)=\mu(E)\) (translation invariance).
The Vitali Non-Measurability Theorem demonstrates that no such function exists. Indeed the closest mathematical generalization of mass is σ-additivity which gives rise to the Lebesgue measure, however not all sets have such a measure (for example, a Vitali set).
To overcome this problem we turn to some standard approaches for mathematicians, either (1) weaken the criteria; or (2) restrict the domain of the function to the family of well-behaved sets. Both options are valid and we will see that (1) leads us to the concept of outer-measure, and (2) leads to the concept of measure spaces and measures.
Let us consider how we might restrict the domain of the function and what kind of family of subsets we should restrict to. The answer is a non-empty collection of subsets that is both closed under compliments and closed under countable unions (i.e. sets that preserve the properties we wish our measure to have) also known as a \(\sigma\)-algebra, a specific type of algebra.
1.2 Algebra over Sets
1.3 σ-Algebra
1.4 Measure
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