Metric Space

A set \(X\) with a distance function \(d: X \times X \rightarrow \mathbb{R}^+\) that satisfies the following:

• Separation: \(d(a,b) = 0 \) if and only if \(a = b\).
• Symmetry: \(d(a,b) = d(b,a)\) for any pair of points \(a, b \in X\).
• Trianly Inequality\(d(a,b) \leq d(a,c) + d(c,b)\) for any three points \(a,b,c \in X\).

Topological Space

A topological space is a pair \((X, T)\) where \(T \subset \mathcal{P}(X)\) (called a topology on \(X\)) that satisfies the following:

• \(X\) and \(\emptyset\) are in \(T\)
• An arbitrary union of sets in \(T\) is also in \(T\)
• A finite intersection of sets in \(T\) is in \(T\)

Sets in \(T\) are called open. Compliments of sets in \(T\) are called closed.

If \(d\) is a metric on \(X\) then sets of the form \(B_\epsilon(x) = \{y \in X | d(x,y) < \epsilon\}\) generate a topology in which \(B_\epsilon(x)\) is the ball of radius \(\epsilon\) around \(x.\)

Measure Space

Let \(X\) be a set with a \(\sigma\)-algebra \(\mathcal{A}\). That is, \(\mathcal{A}\) is a subset of \(\mathcal{P}(X)\) that satisfies the following:

• \(X \in \mathcal{A}\)
• If \(A \in \mathcal{A}\) then \(A^C \in \mathcal{A}\)
• \(\mathcal{A}\) is closed under finite unions

A measure on \((X,\mathcal{A})\) is a function \(\mu: \mathcal{A} \rightarrow \mathbb{R}^+\) such that

• \(\mu(\emptyset) = 0\), and
• If \(\{ A_i \}_{i=1}^\infty\) is a countable set of pairwise disjoing functions, then \(\mu(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \mu(A_i) .\)

The triple \((X, \mathcal{A}, \mu)\) is a measure space.

Borel and Lebeque Measure on \(\mathbb{R}\)

The Borel measure is the measure on the \(\sigma\)-algebra generated by open sets on \(\mathbb{R}\) such that \(\mu((a,b)) = b - a\) for any open interval \((a,b) \subset \mathbb{R}\).

The Lebesgue measure is the completion of the Borel measure, meaning if any set \(A\) has measure \(0\) then every subset of \(A\) is measurable and also has measure \(0.\)

Not every subset of \(\mathbb{R}\) is Lebesgue measurable

The classic example is to create equivalence classes \(a \equiv b\) if \(a - b \in \mathbb{Q}\). Choose one representative from every class in \([0,1)\) and create a set \(A\). For all rationals in \(q \in [0,1)\) we can create a disjoint class \(A_q = \{z : z = a + q\) or \(z = a + q - 1, 0 \leq z < 1,\) for all \(a \in A\}.\)

Every class \(A_q\) must have the same measure. They are a countable collection of disjoing sets, so, \(1 = \mu([0,1)) = \mu(\bigcup_{q \in \mathbb{Q}}A_q) = \sum_{q \in \mathbb{Q}}\mu(A_q).\)

If \(\mu(A) = 0\) we get \(1 = 0\) but if \(\mu(A) > 0\) we get \(1 = \infty,\) so \(A\) cannot be measurable.

Continuous and Measurable Functions

Let \((X,T)\) and \((Y,U)\) be topolotical spacted. Then function \(f:X \rightarrow Y\) is continuous if \(f^{-1}(A) \in T\) for every \(A \in Y.\)

Let \((X,\mathbb{A},\mu)\) be a measure space and \(Y\) have \(\sigma\)-algebra \(\mathbb{B}.\) The function \(f: X \rightarrow Y\) is measurable if for every \(B \in \mathbb{B},\) \(f^{-1}(B) \in \mathbb{A}.\)

Lebesgue Integral

An \(L^+\) simple function is a function of the form \(\sum_{i=1}^\infty a_i 1_{A_i}\) where \(a_i \in \mathbb{R}^+\) and \(A_i \in \mathbb{A}\).

The Lebesgue Intebral of a function \(f \in L^+\) is \(\int f d\mu = \sum_{i=1}^\infty a_i \mu(A_i)\)

For a measurable function \(f: X \rightarrow \mathbb{R}^+,\) \(\int f d\mu = sup\left\{ \int \phi d\mu : \phi \leq f, \phi \in L^+ \right\}.\)

For any function \(f,\) break up \(f\) into \(f^+\) where \(f\) maps to positive values and \(f^-\) where \(f\) maps to negative values. Then \(\int f d\mu = \int f^+ d\mu - \int |f^-| d\mu.\)

Probability Measures

A probably measures is a measure \(P\) on a set \(X\) such that \(P(X) = 1.\) If \(f\) is a measurable function \(f: X \rightarrow \mathbb{R},\) then \(\int f dP = E[f].\)