Borel algebra
If A and B are any two sets, let \(A- B\) be the set of all elements in A that are not in B (This does not imply \(B\subset A\)).
A family \(\mathcal{R}\) of sets is called a ring if \(A,B \in \mathcal{R}\) implies
\[ A\cup B \in \mathcal{R} \],
. \[ A-B \in \mathcal{R} \].
Notice that the conditions imply \(A \cap B \in \mathcal{R}\) if its a ring.
A ring is called a \(\sigma\) ring if
\[ \cup_{n=1}^\infty A_n \in \mathcal{R} \],
for any countable sequence of sets \(A_n \in \mathcal{R}\).
If this condition is fulfilled, then
\[ \cap_{n=1}^\infty A_n \in \mathcal{R} \]
also holds.
A set function \(\phi\) defined on \(\mathcal{R}\) assigns to every element of a number \(\phi(A)\) of the extended number system. \(\phi\) is additive if \(A \cap B = 0\) implies
\[ \phi(A\cup B) = \phi(A)+ \phi(B) \],
and it is countably additive if \(A_i \cap A_j = 0\) (\(i \neq j\)) implies
\[ \phi(\cap^\infty_{n=1}A_n) = \sum^\infty_{n=1}\phi(A_n). \]
Notice that nonnegative additive set functions are monotonic by inclusion. In addition, for additive set functions,
\[ \phi(A-B) = \phi(A) -\phi(B) \]
if \(B\subseteq A\) and \(|\phi B| < +\infty\).
Suppose \(\phi\) is countably additive on a ring \(\mathcal{R}\). If \(A_n \in \mathcal{R}\) and \(A_n\subseteq A_{n+1}\),\(A \in \mathcal{R}\) and
\[ A = \cup_{n=1}^\infty A_n, \]
Then as \(n\to \infty\),
\[ \phi(A_n) \to \phi(A) \]
Lebesgue measure
An elementary set is a union of a finite element of intervals. The family of all elementary subsets of \(\mathbb{R}^p\) is denoted \(\mathcal{E}\).
The following properties may be verified,
- \(\mathcal{E}\) is a ring but not a $σ$-ring.
- \(m\) defined by \(m([a,b]) = \Pi_{i=1}^p (b_i-a_i)\) is well defined and additive on \(\mathcal{E}\).
A nonnegative additive set function is regular if for every \(A \in \mathcal{E}\) and to every \(\varepsilon > 0\) there exist sets \(F,G \in \mathcal{E}\) such that \(F\) is closed, \(G\) is open, \(F \subset A \subset G\), and
\[ \phi(G) - \varepsilon \leq \phi(A) \leq \phi(F)+\varepsilon \]
\(m\) is regular, \(\mu(a,b) = \alpha(b) - \alpha(a)\) where \(\alpha\) is monotonically increasing is regular on the real line.
Every regular set function on \(\mathcal{E}\) can be extended to a countably additive set function on a \(\sigma\) -ring which contains \(\mathcal{E}\).
Let \(\mu\) be additive, regular, nonnegative and finite on \(\mathcal{E}\). Consider countable coverings of any set \(E\subseteq \mathbb{R}^p\) by open elementary sets \(A_n\):
\[ E \subseteq \cup_{n=1}^\infty A_n \]
Define \[ \mu^*(E) = \inf \sum_{n=1}^\infty \mu(A_n), \] the inf being taken over all countable coverings of \(E\) by open elementary sets. \(\mu^*(E)\) is called the outer measure of \(E\), corresponding to \(\mu\).
Properties of an outer measure.
- The outer measure is nonnegative for all \(E\).
- It is monotonic by inclusion.
- Its restriction to elementary set is equivalent to the original measure.
- Subadditivity: If \(E = \cup_1^\infty E_n\), then
\[ \mu^*(E) \leq \sum_{n=1}^\infty \mu^*(E_n) \]
Distances between sets
For any \(A,B\subseteq\mathbb{R}^p\), we define the symmetric difference and the distance functions
\[ S(A,B) = (A-B)\cup(B-A) \], \[ d(A,B) = \mu^*(S(A,B)). \]
We write \(A_n \to A\) if \(\lim_{n\to\infty} d(A,A_n) = 0\).
- If there is a sequence of elementary sets converging to \(A\), we say its finitely \(\mu\) measurable
and write \(A \in \mathfrak{M}_F(\mu)\).
- If \(A\) is the union of a countable collection of finitely \(\mu\) measurable sets, then \(A\) is \(\mu\) measureable and write \(A \in \mathfrak{M}(\mu)\).
Properties of the symmetric difference and the distance function.
- \(S(A,B)\subseteq S(A,C)\cup S(C,B)\).
- \(S(A_1 \cup A_2,B_1\cup B_2), S(A_1 \cap A_2,B_1\cap B_2), S(A_1 - A_2,B_1- B_2) \subseteq S(A_1,B_1) \cup S(A_2,B_2)\)
The distance function follow analagous relations with the relevant set operations becoming the appropriate arithmetic operations.
There is also the property
\[ \vert \mu^*(A) - \mu^*(B) | \leq d(A,B) \]
if at least one of \(\mu^*(A),\mu^*(B)\) is finite.
The distance function is almost a true metric but different sets can have zero distance. Thus we may instead define equivalence in terms of vanishing distance. This equivalence relation makes \(\mathfrak{M}_F(\mu)\) be the closure of \(\mathcal{E}\).j
\(\mathfrak{M}(\mu)\) is a $σ$-ring and \(\mu^*\) is countably additive on \(\mathfrak{M}(\mu)\).
The extended set function is called a measure, and the Lebesgue measure on \(\mathbb{R}^p\) is thus the special case \(\mu = m\).
Borel sets
A borel set is a set that can be obtained by a countable number of unions,intersectoins or complements starting from open sets. The collection of all Borel sets is a \(\sigma\) -ring, in fact the smallest one which contains all open sets. An element of a borel set is \(\mu\) measureable. Every \(\mu\) measurable set is the union of a Borel set and a set of measure zero.
Measure spaces
A set \(X\) is a measure space if there exists a \(\sigma\) ring \(\mathfrak{M}\) of subsets of \(X\) (the measurable sets) and a non-negative countably additive measure defined on \(\mathfrak{M}\). If, in adddition, \(X \in \mathfrak{M}\), then \(X\) is said to be a measurable space.
Measurable functions
A function is measurable if the set \(\{x | f(x) > a\}\) is measurable for every real \(a\). Equality can be included.
Making more measurable functions.
- If \(f\) is measurable, then \(|f|\) is measurable.
- Let \(\{f_n\}\) be a sequence of measurable functions. For \(x \in X\), put
\[ g(x) = \sup f_n(x) \]
\[ h(x) = \lim_{n\to\infty}\sup f_n(x) \], Then \(g\) and \(h\) are measurable.
This results in the following corollaries,
- If \(f\) and \(g\) are measurable, then \(\max(f,g)\) and \(\min(f,g)\) are measurable
- The limit of a convergent sequence of measurable functions is measurable.
Common binary operations are also preserve measurability.
Let \(f\) and \(g\) be measurable real-valued functions defiend on \(X\), let \(F\) be real and continuous on \(\mathbb{R}^2\). Then \(F(f(x),g(x))\) is measurable.
Thus common operations of analysis can be applied to measurable functions to give measurable functions. An example where measurability doesn’t carry over is \(h(x) = f(g(x))\), where \(f\) is measurable and \(g\) is continuous, but \(h\) is not necessarily measurable. Interestingly, we were able to define measurable functions without referencing any particular measure. Thus, the class of measurable functions on \(X\) depends on the \(\sigma\) ring \(\mathfrak{M}\).
Simple functions
Let \(s\) be a real-valued function defined on \(X\). If the range of \(s\) is finite, then \(s\) is a simple function. Let \(E \subseteq X\) and put define \(K_E(x) = 1\) when \(x\in E\) and zero otherwise as the characteristic function of \(E\). Denoting \(E_i = s^{-1}(\{c_i\})\) where \(c_i\) make up the range of \(s\), we can decompose any simple function into a linear combinations of characteristic functions, \[ s = \sum_{i=1}^n c_i K_{E_i}. \]
\(s\) is measurable iff \(E_i\) are measurable.
This allows us to approximate every function by simple functions.
Let \(f\) be a real function on \(X\). There exists a sequence \(\{s_n\}\) of simple funcitons such that \(s_n(x) \to f(x)\) as \(n \to \infty\), for every \(x \in X\). If \(f\) is measurable, \(\{s_n\}\) may be chosen to be a sequence of measurable functions. If \(f\geq 0\), \(\{s_n\}\) may be chosen to be a monotonically increasing sequence.
\(s_n\) can be constructed as
\[ s_n = \sum_{i=1}^{n2^n}\frac{i-1}{2^n} K_{E_{ni}} + n K_{F_n} \]
for non-negative \(f\). For general functions, the function can be split into the difference between its positive and negative components. The sequence converges uniformly to \(f\) if \(f\) is bounded.
Lebesgue Integration
Suppose \((x \in X, c_i > 0)\) \[ s(x) = \sum_{i=1}^n c_i K_{E_i}(x) \] is measurable and \(E \in \mathfrak{M}\). We define
\[ I_E(s) = \sum_{i=1}^n c_i \mu(E \cap E_i). \] If \(f\) is measurable and non-negative, we define
\[ \int_E f \dd{\mu} = \sup I_E(s), \]
where the sup is taken over all measurable simple functions \(s\) such that \(0 \leq s \leq f\).
The integral may have the value \(+ \infty\). If the integral of each of the positive and negative components of a function is finite, then \(f\) is said to be integrable, or \(f \in \mathcal{L}(\mu)\). Integrability carries over to subsets.
Suppose \(f\) is measurable and non-negative on \(X\). For \(A \in \mathfrak{M}\), define
\[ \phi(A) = \int_A f\dd{\mu}. \]
Then \(\phi\) is countably additive on \(\mathfrak{M}\). The same conclusion holds for any \(f \in \mathcal{L}(\mu)\).
Suppose \(E\in\mathfrak{M}\). Let \(\{f_n\}\) be a sequence of measurable functions such that \(0 \leq f_1(x) \leq f_2(x) \leq \ldots\). Let \(f\) be defined by \(f_n(x) \to f(x)\) as \(n\to\infty\). Then
\[ \int_E f_n \dd{\mu} \to \int_E f\dd{\mu} \]
A corollary of the result is that the lebesgue integral and summation of a sequence of measurable functions commute.
Suppose \(E \in \mathfrak{M}\). If \(\{f_n\}\) is a sequence of non-negative measurable functions and
\[ f(x) = \lim_{n\to\infty}\inf f_n(x), \]
then
\[ \int_E f \dd{\mu} \leq \lim_{n\to\infty} \inf \int_E f_n \dd{\mu} \]
Suppose \(E \in \mathfrak{M}\). Let \(\{f_n\}\) be a sequence of measurable functions such that \[ f_n(x) \to f(x) \] as \(n\to\infty\). If there exists a function \(g \in \mathcal{L}(\mu)\) such that
\[ \vert f_n(x) | \leq g(x) \]
for all \(n\), then,
\[ \lim_{n\to\infty} \int_E f_n \dd{\mu} = \int_E f \dd{\mu} \]
In summary, the limits of measurable functions are always measurable, whereas the limits of Riemann-integrable functions may fail to be Riemann-integrable.