1 Measure Theory
Our primary aim in measure theory is to define the concept of measurability and ultimately assign a measure to measurable sets. This will be useful when we define the Lebesgue integral, and furthermore, this has applications in probability theory where measurable sets may be considered as those events for which a probability can be calculated.
1.1 Measure Spaces
We first define a collection of subsets of \(\Omega\) with certain desirable properties.
For general set \(\Omega\) a collection \(\mathcal{F}\subset 2^\Omega\) is called a \(\sigma\)-algebra in \(\Omega\) iff it satisfies the following conditions: 1. Contains the whole space. \(\Omega \in \mathcal{F}\). 2. Closed under compliments. \(A\in\mathcal{F}\implies A^c\in \mathcal{F}\). 3. Closed under countable unions. \(\{ A_{n} \}\subset \mathcal{F}\) then \[ \bigcup_{n}A_{n}\in\mathcal{F}. \] The tuple \((\Omega, \mathcal{F})\) consisting of a set \(\Omega\) and a \(\sigma\)-algebra \(\mathcal{F}\) on \(\Omega\) is called a measurable space.
We note that this definition automatically implies: 1. Closure with respect to countable intersections via De Morgan. 2. \(\emptyset \in \mathcal{F}\).
We contrast the definition of a \(\sigma\)-algebra with that of a topology.
For general set \(X\) a collection \(\mathcal{T}\subset 2^X\) is called a topology on \(\Omega\) iff it satisfies the following conditions: 1. Contains the whole space and the empty set. \(\Omega, \emptyset \in \mathcal{T}\). 2. Closed under finite intersections. \(\{ U_{n} \}\subset \mathcal{T}\) then \[ \bigcap_{j=1}^n U_{j}\in \mathcal{T}. \] 3. Closed under arbitrary unions. \(\{ U_{\alpha} \}\subset \mathcal{T}\) for \(\alpha \in I\) (for potentially uncountable \(I\)) then \[ \bigcup_{\alpha \in I}U_{\alpha} \in\mathcal{T}. \] The tuple \((X, \mathcal{T})\) consisting of a set \(X\) and a topology \(\mathcal{T}\) on \(X\) is called a topological space.