2 Measures
2.1 Properties
2.1.1 Definition
A measure \mu : \mathcal E \rightarrow \bar{\mathbb R_+} is a set function satisfying:
- Non-negativity: \mu(A) \geq 0 \forall A
- Countable additivity: for a countable sequence of disjoint sets (A_i), \mu(\cup_i A_i) = \sum_i \mu(A_i)
2.1.2 Uniqueness
A finite measure is uniquely determined by its values on a p-system generating the sigma algebra.
If \mu, \nu are measures on (E , \mathcal E) such that \mu(E) = \nu(E) < \infty, and \mu(A) = \nu(A) \forall A \in \mathcal C such that \sigma C = \mathcal E and A, B \in \mathcal C \implies A\cap B \in \mathcal C.
The proof is a straightforward application of the monotone class theorem.
A corollary of this is that CDFs uniquely characterise probability measures on (\bar{\mathbb R}, \mathcal B(\bar{\mathbb R})).
This can be extended to \sigma-finite measures, with the \mu(E) = \nu(E) < \infty requirement replaced by the condition that \mathcal C contains a countable measurable partition (E_n) of E such that \mu(E_n) = \nu(E_n) < \infty for each n.
2.1.3 Zero on null set
Proof: Take A_i = \emptyset. We get \cup_i A_i = \emptyset, thus \mu(\emptyset) = \sum_{i=1}^\infty \mu(\emptyset), implying \mu(\emptyset) = 0.
2.1.4 Monotonicity
A \subset B \implies \mathbb \mu(A) \leq \mathbb \mu(B)
2.1.5 Sequential continuity from below:
For a monotone increasing sequence of sets A_n \nearrow A, \mu (A_n) \nearrow \mu(A).
Proof
Define B_1 = A_1, B_i = A_i \setminus (\cup_{j=1}^{i-1} A_j). (B_i) is a countable sequence of disjoint sets such that A_n = \cup_{i =1}^n B_i, i.e. \mu(A_n) = \sum_{i=1}^n \mu(B_i) . \mu(A) = \mu(\cup_i B_i) = \sum_i \mu(B_i) = \lim_{n \to \infty} \sum_{i=1}^n \mu(B_i) = \lim_{n \to \infty} \mu(A_n)
And the sequence is increasing because \mu(A_n) = \mu(A_{n-1}) + \mu (B_n) \geq \mu(A_{n-1}).2.1.6 Sequential continuity from above
For a monotone decreasing sequence of sets A_n \searrow A, if for some i \mu(A_i) < \infty then \mu (A_n) \searrow \mu(A)
2.1.7 Countable subadditivity
\mu(\cup_{n} A_n) \leq \sum_n \mu(A_n)
2.1.8 Closed under linear combination
If (\mu_n) is a sequence of measures on some measurable space and (a_n)\subset \mathbb R , \sum_n a_n \mu_n is also a valid measure.
2.2 Types
Finite \implies \sigma finite \implies \Sigma-finite
2.2.1 Finite measures
\mu(E) < \infty
2.2.2 Sigma-finite measures
A measure \mu defined on (E, \mathcal E) is \sigma-finite iff there exists a countable partition (E_i) (\cup_i E_i = E) where \mu(E_i) < \infty.
2.2.3 Sum finite
A measure \mu defined on (E, \mathcal E) is \Sigma-finite iff there exists a countable sequence of finite measures (\nu_n) such that \mu = \sum_n \nu_n.
2.3 Constructing measures
Any measure in a standard measurable space can be expressed as a transformation of the Lebesgue measure on \mathbb R_+.
2.3.1 Pushforward measures
Let (E, \mathcal E), (F, \mathcal F) be measurable spaces. Let h: E \to F be measurable and \nu be a measure on E. The image or pushforward measure \nu \circ h^{-1}: \mathcal F_+ \to \bar{\mathbb R_+} is defined as:
\nu \circ h^{-1} (A) = \nu (h^{-1} A) \qquad \forall A \in \mathcal F
And the corresponding Lebesgue integral for a measurable function f \in \mathcal F_+ is:
\nu \circ h^{-1}[f] = \int f(y) \nu \circ h^{-1} (dy) = \int f(h(x)) \nu(dx) = \nu[f \circ h]
A special case (change of variable) when E = F= \mathbb R and \nu = \lambda Lebesgue measure, and h is strictly increasing and h^{-1} is differentiable is: \int f(y) \frac{dh^{-1}(y)}{dy} dy = \int f(h(x)) dx
2.3.2 Density integral measures
Let p \in \mathcal E_+, we can create a new measure as:
\nu (A) = \int_A p(x) \mu(dx) = \mu[p \mathbb 1_A]
The corresponding Lebesgue integral is:
\nu[f] = \int f(x) p(x) \mu(dx) = \mu[pf]