\( f_n \text{ is uniformly bounded } \iff \exists M ( |f_n (x)| \leq M ) \)
Property for sequence of functions such that for each domain element epsilon pair, there exists a single \(\delta_x\) that proves continuity at that point for all indexed functions
\( f_n \text{ is equicontinuous } \iff \forall x \forall \varepsilon ( \exists \delta_{x} \gt 0 ( |x - y| \lt \delta_{x} \implies |f_n(x)-f_n(y)| \lt \varepsilon ) ) \)
Property for sequence of functions such that for each epsilon, there exists a single \(\delta\) that proves uniform continuity for all indexed functions
\( f_n \text{ is uniformly equicontinuous } \iff \forall \varepsilon ( \exists \delta ( |x - y| \lt \delta \implies |f_n(x)-f_n(y)| \lt \varepsilon ) ) \)
Theorem asserting necessary and sufficient conditions for compactness for spaces of continuous functions on closed and bounded real intervals. It is similar in spirit to BWT, however instead of real space, it discusses the space of continuous real functions.
Theorem asserting that every IVP for continuous, explicit first-order differential equation has a solution.
An extension of the Riemann integral that applies some function \(\phi\) to the domain elements (
\( f \text{ is Riemann-Stieltjes integrable on } [a,b] \iff \exists \forall \varepsilon( \exists \delta: ( \xi \in [x_{i-1} , x_i] \land |x_i - x_{i-1}| \lt \delta \implies |\sum^{n}_{i=1} f(\xi)\Delta \phi_{i} - L| \lt \varepsilon ))\)
\( \displaystyle \text{RS} \int^{b}_{a} f(x) d\phi = L\)
\( \displaystyle f \text{ is continuous on } [a,b] \land \phi \text{ is a step function} \implies \text{RS}\int_{a}^{b} f(x)d\phi = \sum_{i=1}^{n} f(x_i) [\phi(x_{i}^{+}) -\phi(x_{i}^{-}) ]\)
\( \displaystyle f \text{ is continuous on } [a,b] \land \phi \in C([a,b]) \implies \text{RS}\int_{a}^{b} f(x)d\phi = \int_{a}^{b} f(x) \phi'(x)dx\)
\( \displaystyle f,\phi \text{ are bounded on } [a,b] \land \exists \text{RS}\int^{b}_{a} f(x)d\phi \implies \exists \text{RS}\int_{a}^{b} \phi (x)df = f(b)\phi(b) - f(a)\phi(a) - \text{RS}\int_{a}^{b} f(x) d \phi\)
Space of sets that is closed under set minus and finite unions
\(\mathcal{F} \text{ is a ring of sets on } X \iff\)
Space of sets that is closed under complement and finite unions
\(\mathcal{A} \text{ is an algebra of sets on } X \iff\)
Set in a topological space formed by countable unions, countable intersections, and complements
\(X=\mathbb{R} \implies \mathfrak{B} = \{ A \subset \mathbb{R} : A= ( \bigcup^{\infty}_{i=1} A_n) \cup ( \bigcup^{\infty}_{i=1} \overline{B_n} ) \}\)
Algebra of sets with closure under countable unions
\(\Sigma \subseteq \mathcal{P}(X) \text{ is a } \sigma \text{-algebra on } X \iff\)
On a topological space \(X\), the \(\sigma\)-algebra \(\mathcal{B}\) made from the intersection of all \(\sigma\)-algebra on some space \(X\) using open interval.
\(\mathcal{B} = \bigcap_{n} \Sigma_{n}\)
It is called the smallest \(\sigma\)-algebra in the sense that all other \(\sigma\)-algebrae on \(X\) are supersets of \(\mathcal{B}\)
Set function \(\mu : \Sigma \to [0,\infty) \) that applies some notion of 'size' to a set
\(\mu \text{ is a measure on } (X,\Sigma ) \iff \)
\(\Sigma\) is technically a type of set called a \(\sigma\)-algebra, but until this concept is discussed, assume it is just the domain of meaurable sets.
Set function \(\mu^{*} : \mathcal{P}(\mathbb{R}) \to [0,\infty)\) that resembles a weaker variant of a measure function that 'approximates from above', analogue to an upper Riemann sum. it is defined such that (countable sub-additivity)
\( \mu : \Sigma \to \mathbb{R}\cup \{ -\infty , + \infty \} \text{ is a signed measure on } (X, \Sigma) \iff \)
Notion of a measure on an algebra of sets, which guarantees the existence of a measure when a \(\sigma\)-algebra is generated by this algebra of sets. This is the essence of the Caratheodory Extension Theorem, and it is necessary since an algebra of sets doesn't guarantee the inclusion of countable unions of its sets
\(\mu \text{ is a premeasure on } (X,\mathcal{A}) \iff \)
\(I = [a,b] \text{ is an interval } \implies \ell (I) = b-a \)
Property of a measure or signed measure such that there is no set of infinite measure
\( \mu \text{ is finite } \iff \sup_{\sigma \in \Sigma} |\mu(\sigma)| \lt \infty\)
\( \mu \text{ is finite } \iff \sup_{\sigma \in \Sigma} |\mu(\sigma)| \lt + \infty\)
\( \mu \text{ is } \sigma \text{-finite on } X \iff \exists X_n : \mu (X_n)\lt \infty \land \bigcup^{\infty}_{n=1} X_n = X \)
\(\lambda^{*} :\mathcal{P}(\mathbb{R}) \to [0,\infty ) \)
\(\lambda^{*}(S) = \inf \{ \sum_{k} \ell (I_k) : S \subseteq \bigcup_{k} I_k \}\)
\( S_i \text{ are disjoint } \implies \lambda^{*} ( \bigcup_{n=1}^{\infty} S_n ) \leq \sum_{n=1}^{\infty} \lambda^{*} (S_{n}) \)
\(\mu^{*}(S_n) \leq \sum_{k} U_k - \frac{\varepsilon}{2^n}\)
\(\mu^{*}(S) \leq \sum_{n} \sum_{k} U_k - \frac{\varepsilon}{2^n} \leq \sum_{k} U_k - \varepsilon\)
\(\lambda_{*}(S) = \sup \{ \sum_{k} \ell (I_k) : \bigcup_{k} I_k \subseteq S \}\)
Set that can be measured by Legesbue's definition, \(\mathcal{L}\) represente the set of all Lebesgue measurable sets and \(\lambda=\lambda^{*}\) is the Lebesgue measure function
\(S \text{ is Lebesgue mesurable } \iff S \in \mathcal{L} \)
\(S \text{ is Lebesgue mesurable } \iff \lambda^{*}(S) = \lambda_{*} (S) = \lambda (S) \)
\(S \text{ is Lebesgue mesurable } \iff \forall A \subseteq \mathbb{R}, \lambda^{*}(A) = \lambda^{*} (A \cap S) + \lambda^{*} (A \cap \overline{S})\)
\(S \text{ is Lebesgue mesurable } \iff \forall A \subseteq \mathbb{R}, \lambda^{*}(A) \geq \lambda^{*} (A \cap S) + \lambda^{*} (A \cap \overline{S})\) (Since the case where the measure is less than is guaranteed by a previous lemma)
\( E \text{ is a positive set } \iff \forall S \subseteq E, \mu (S) \geq 0 \)
Class of sets \(V \subset [0,1]\) such that for each \(r \in \mathbb{R}\), there is a unique element \(v \in V\), that \(v-r \in \mathbb{Q}\). Such a set is uncountable and Lebesgue non-measurable
\(V \subset [0,1] \text{ is a Vitali set } \iff \forall r \in \mathbb{R} ( \exists! v \in V ( v-r \in \mathbb{Q} ) ) \)
Theorem asserting that the Lebesgue measurable real sets are a strict subset of the real powerset; there are sets that are not Lebesgue measurable
\(\mathcal{L}(\mathbb{R}) \subset \mathcal{P}(\mathbb{R})\)
By way of contradiction, assume a Vitali set \(V\) is Lebesgue measurable.
Let \( \{r_i\}^{\infty}_{i=1}\) be an enumeration of \(\mathbb{Q}\).
\(V \text{ is a Vitali set } \implies \forall x \in \mathbb{R} ( \exists! v \in V ( \exists! r \in \mathbb{Q} (x = v + r))) \implies (V+r_i) \cap (V+r_j) = \emptyset \)
By translation invariance, \(\lambda(V+r_i) = \lambda (V+r_j)\).
By nonnegativity, \(\lambda(V) \geq 0 \).
\( \lambda (V) \gt 0\), otherwise \(\lambda( \mathbb{R} ) = \lambda ( \bigcup^{\infty}_{i=1} V +r_i ) = \sum^{\infty}_{i=1} \lambda ( V +r_i\) = 0 \).
\( \lambda ( [0,2] ) \geq \lambda ( \bigcup^{\infty} _{i=1} \{ V+r_i : r_i \in [0,1] \} ) = \sum^{\infty} _{i=1} \lambda ( \{ V+r_i : r_i \in [0,1] \} ) = \infty \)
However \(\lambda ( [0,2] ) =2\), therefore a contradiction is encountered. ■
Set that is defined in the domain of the measure function
Linear combination of characteristic functions
\(f \text{ is simple } \iff \exists a_n, A_n : f(x) = \sum^{k}_{n=1} a_n \chi_{A_n}(x) \)
\(f \text{ is measurable } \iff \forall S : S \text{ is open , }f^{-1}(S) \text{ is measurable} \)
\(f \text{ is measurable } \iff \forall a \in \mathbb{R} \{x : f(x) \gt a\} \text{ is measurable} \)
\(f: X \to Y \text{ is continuous} \land X \text{ is measurable } \implies f \text{ is measurable} \) (proof by topological definition of continuity)
\(S \text{ is measurable } \iff \chi_{S} \text{ is measurable } \)
\(\forall n, f_n \text{ is measurable } \land \lim_{n \to \infty} f_n = f \text{ pointwise } \implies f \text{ is measurable } \)
\(f,g \text{ are measurable } \land c \text{ is a constant} \implies \)
Condition of a property holding for a whole universe of discourse with the exception of a measure-zero subset
\(P(X) \text{ holds almost everywhere } \iff P(X \setminus N ) \land \mu (N) = 0\)
Smallest supremum of a function's codomain on a set minus a measure-zero subset (smallest supremum almost everywhere)
\( \text{ess sup} (U) = \inf \{ a \in X : \mu\{ x : f(x) \geq a \}= 0 \} \)
Theorem asserting that measurable sequences of functions that converge pointwise almost everywhere on a space converge uniformly almost everywhere.
The proof works by deleting an arbitrarily small subset of points where uniform convergence does not hold.
\( ( \forall n ( f_{n} \text{ is measurable } ) \land \lim_{n \to \infty} f_n = f \text{ pointwise a.e. on } E ) \implies \forall \varepsilon \gt 0 , \exists F \subset E \)
\(E_{n}^{m} = \bigcap^{\infty}_{i=n} \{ x : | f_i (x) - f(x) | \lt \frac{1}{m} \} \) represents points that are within \(\frac{1}{m}\) from sequence indexes \(i \geq n \). Note that:
One can show through set algebra that \(E \setminus ( E \setminus \bigcup^{\infty}_{m=1} E_{N(m)}^{m} ) = \bigcup^{\infty}_{m=1} E^{m}_{N(m)} \) which represents the set of all uniformly convergent points of \(f_n\), proving the theorem
\( f_{n} \to f\text{ in the Egorov sense } \iff \forall \delta \gt 0(\exists E \subset [a,b] : E \text{ is measurable } \land \mu(E) \lt \delta \land \lim_{n \to \infty} f_n = f \text{ uniformly on } [a,b] \setminus E) \)
\( f \text{ is measurable on } E \implies \exists F \subset E \)
Letting \( f_n \) be a sequence of measurable functions, convergence in measure extends the idea of convergence
\( f_n \to f \text{ in measure } \iff \forall \varepsilon (\lim_{n \to \infty} \mu( \{ x: |f_n(x) - f(x)| \geq \varepsilon \}) )=0 ) \)
\( f_n \text{ is Cauchy in measure } \iff \forall \varepsilon ( \exists N : n,m \geq N \implies \mu( \{ x: |f_n(x) - f_m(x)| \geq \varepsilon \}) ) \lt \delta ) \)
\( \lim_{n \to \infty } f_n = f\text{ pointwise almost everywhere } \implies f_n \to f \text{ in measure } \)
\( f_n \text{ is Cauchy in measure } \iff \lim_{n \to \infty} f_n = f \text{ in measure } \)
\( f_n \text{ is Cauchy in measure } \implies f_n \to f \text{ in the Egorov sense } \)
Paradox that says that two bounded sets in \(\mathbb{R}^3\) Euclidean space can be finitely partitioned into sets such that each partition in one set is congruent to one in the other
Theorem stating that on a signed measure space, the space can be decomposed as one positive set and one negative set, splitting one signed measure space into two measure spaces
\( (X, \Sigma, \mu) \text{ is a signed measure space } \implies \)
\(\mu(E) = \mu^{+}(E) - \mu^{-}(E)\)
Property of a space such that all null sets of \(\nu\) are null sets of \(\mu\), but not necessarily the converse
\( \nu \ll \mu \iff (\mu (E) = 0 \implies \nu(E) = 0 ) \)
Property of a space such that \(\nu, \mu\) share all of their null sets
\( \nu \sim \mu \iff ( \nu \ll \mu \land \mu \ll \nu ) \)
Property of a space such that when partitioned, the measures \(\nu, \mu\) of opposite partitions are both zero
\(\nu \perp \mu \iff ( A\cup B= X \land A \cap B = \emptyset \land \nu(A) = \mu (B)=0 ) \)
Theorem asserting that for a measure space, absolute continuity is the sufficient condition for a measure function for \(\nu\) to be defined in terms of integration of some unique function with respect to \(\mu\)
\( \mu ,\nu \text{ are } \sigma \text{-finite measures } \land \nu \ll \mu \implies \)
This theorem asserts the sufficient conditions for a change-of-measure, hence the followin derivative notation becomes convenient
\(\displaystyle \frac{d \nu}{d \mu} : \int_E \frac{d\nu}{d\mu} d\mu = \int_E d\nu \)
Weaker variant of algebra of sets that has semi-closure under finite unions; every semi-algebra has a smallest algebra of sets that contains it, and such a set can have a meassure upon it. The cartesian product of sigma algebras is a semi algebra, hence working with this is important for generalising integration to several dimensions
All semi-algebras have some smallest algebra that contains it
Smallest \(\sigma\)-algebra containing the semi-algebra of the product of two measurable spaces. This may be denoted as \(\Sigma_1 \otimes \Sigma_2\)\)
Theorem asserting that a premeasure \(\mu\) on an algebra \(\mathcal{A}\) guarantees a unique measure \( \mu' =\mu\) on the \(\sigma\)-algebra generated by \(\mathcal{A}\). This is useful since on a \(\sigma\)-algebra, the existence of countable unions of sets is guaranteed to be included in the \(\sigma\)-algebra. Since the cartesian product of sigma algebras is a semi-algebra, this theorem is necessary for multivariate integration
\(\mu \text{ is a } \sigma \text{-finite premeasure on a algebra of sets } \mathcal{A} \implies \exists ! \sigma \text{-finite measure } \nu ( \forall E \in \Sigma \subseteq \mathcal{A} ( \mu(E) = \nu (E) ) )\)
Measure variant defined on a semi-algebra
\( \Sigma_1 \otimes \Sigma_2 \)
Cartesian product of measurable sets from different measurable spaces
\(A \times B \subseteq X\times Y\ \text{ is a measurable rectangle } \iff ( A \text{ is } (X,\Sigma_X , \mu) \text{-measurable} \land B \text{ is } (Y,\Sigma_Y , \nu) \text{-measurable} ) \)
Set of all countable unions of \(\mathcal{A}\)
Set of all countable intersections of \(\mathcal{A}_{\sigma}\)
Subset of ordered pairs such that for some specific index, all ordered pairs have the same constant value
\( E_x = \{ y \in Y : (x,y) \in E\} \)
Measure on a Borel sigma algebra
\( \mu : \mathcal{B} \to [0,\infty) \text{ is a regular Borel measure } \iff \)
Measurable set with nonnegative measure such that its subsets have idential or zero measure
\(A \text{ is an atom of } \mu \iff\)
Measure that captures the notion of the Dirac delta function using a singular atom \( \{x\} \). It is a regular Borel measure as well as an atomic measure
\( \delta_x (E) = \begin{cases} 1 & x \in E \\ 0 & x \notin E \end{cases}\)
Simple functions with sufficient properties for Lebesgue integral. The weaker Lebesgue intergal for simple function is given below
\( \displaystyle f = \sum^{n}_{i=1} a_i \chi_{A_i} \text{ is an ISF} \iff A_i \text{are disjoint } \land \int f d\lambda = \sum^{n}_{i=1} a_i \lambda (A_i) \lt \infty\)
\( \displaystyle \int f d\lambda = \sum^{n}_{i=1} a_i \lambda (A_i) \)
\( \displaystyle |\int f| \leq \int |f|\)
\( f,g \text{ are ISFs }\implies f+g \text{ is an ISF }\)
\(f \text{ is an ISF } \land c \text{ is a constant} \implies cf \text{ is an ISF }\)
Property of ISFs that converge fast enough such that their convergence is closed under the Lebesgue integral
\( f_n \to f \text{ in mean } \iff \forall \varepsilon ( \exists N : n,m \geq N \implies \int |f_n -f| d\lambda \lt \varepsilon ) \)
\( f_n \text{ is Cauchy in mean } \iff \forall \varepsilon ( \exists N : n,m \geq N \implies \int |f_n -f_m| d\lambda \lt \varepsilon ) \)
\(f \text{ is Lebesgue integrable } \iff f \text{ is finite valued a.e. } \land \exists f_n \text{ of ISFs} : f_n \text{ is Cauchy in mean } \land f_n to f \text{ in measure }\)
\(f \text{ is Lebesgue integrable } \iff \int f d\lambda = \lim_{n \to \infty} \int_{E} f_{n} d\lambda\)
Theorem stating when uniformly bounded measurable sequence of functions may pass a limit through the intergal
\( \implies \lim_{n \to \infty } \int_{E} f_n d \lambda = \int_{E} f d \lambda \)
Theorem stating that all sequences of functions where each term is strictly positive on \(E\) can have the limit passed through the Lebesgue integral
\( \implies \liminf_{n \to \infty } \int_{E} f_n d \lambda = \int_{E} f d \lambda \)
Theorem stating that monotone measurable sequence of functions may pass a limit through the intergal
\( \implies \limsup_{n \to \infty } \int_{E} f_n d \lambda = \int_{E} f d \lambda \)
Theorem stating that sequences of functions absolutely dominated by some integrable function may pass a limit through the intergal
\( \implies \lim_{n \to \infty } \int_{E} | f_n -f | d \lambda = 0 \)
\(f \text{ is Riemann integrable on } [a,b] \implies \)
Theorem stating the following formula for double integrals
\( \iint_{X \times Y} f(x,y) d(x,y) = \int_{X} ( \int_{Y} f(x,y) dy ) dx = \int_{Y} ( \int_{X} f(x,y) dx ) dy \)
Theorem stating the following formula for double integrals
\( f(x,y) \text{ is Lebesgue integrable on } \sigma \text{-finite measure space } (X \times Y,\Sigma_X \otimes \Sigma_Y, \mu \times \nu) \)
\( f_x (y)=f(x,y) \text{ is Lebesgue integrable on } \sigma \text{-finite measure space } (Y,\Sigma_Y, \nu) \)
\( f_y (x)=f(x,y) \text{ is Lebesgue integrable on } \sigma \text{-finite measure space } (X,\Sigma_X, \mu) \)
\( F_1 (x) = \int_Y f_x(y) d\nu \text{ is Lebesgue intergable on } \mu\)
\( F_2 (y) = \int_X f_y(x) d\mu \text{ is Lebesgue intergable on } \nu\)
\( \iint_{X \times Y} f(x,y) d\mu \times \nu = \int_{X} ( \int_{Y} f(x,y) d\nu ) d\mu = \int_{Y} ( \int_{X} f(x,y) d\mu ) d\nu \)
Variant of Fubini's theorem that requires non-negativity on \(X\times Y\) rather than Lebesgue integrability on \(X \times Y\)
\( f(x,y) \geq 0 \text{ on } X \times Y \)
\( f_x (y)=f(x,y) \text{ is Lebesgue integrable on } \sigma \text{-finite measure space } (Y,\Sigma_Y, \nu) \)
\( f_y (x)=f(x,y) \text{ is Lebesgue integrable on } \sigma \text{-finite measure space } (X,\Sigma_X, \mu) \)
\( F_1 (x) = \int_Y f_x(y) d\nu \text{ is Lebesgue intergable on } \mu\)
\( F_2 (y) = \int_X f_y(x) d\mu \text{ is Lebesgue intergable on } \nu\)
\( \iint_{X \times Y} f(x,y) d\mu \times \nu = \int_{X} ( \int_{Y} f(x,y) d\nu ) d\mu = \int_{Y} ( \int_{X} f(x,y) d\mu ) d\nu \)
Ordered pair \( (d,X) \) representing a space \(X\) with a norm function \( d : X \times X \to \mathbb{R}^{+} \) that defines the distance between two values. From this notion, the following constraints apply to metrics:
\( (d,X) \)
\( (d,X) \text{ is a metric space } \iff \forall (x,y),\)
\( (d,X) \text{ is complete } \iff (\forall a_n \subset X , a_n \text{ is Cauchy } \implies \lim_{n \to \infty} a_n \in X) \)
Ordered pair \( ( V, \| \cdot \| ) \) representing a linear space with a norm function. These are therefore "Linear metric spaces".
Complete normed linear space
\( ( V, \| \cdot \| ) \)
Ordered pair \( (V,d) \) representing a linear space with an inner product function (which induces a norm). These are therefore "Linear metric spaces with inner products".
Complete inner product space
\( ( V, \langle \cdot \rangle ) \)
\(L^2 \text{ is a Hilbert space}\)
Complete, inner product space. These spaces always have orthonormal basis'.
\( \| f+ g \|^2 + \| f-g \|^2 = 2 (\|f\|^2 + \|g\|^2 )\)
3-tuple \( ( X, \Sigma, \mu) \) of a \(\sigma\)-algebra, a space \(X\) on which it exists, and some measure function \(\mu\) such that:
Ordered pair \( (X, \Sigma) \) such that:
\( ( \mathbb{R}, \mathcal{L}, \lambda ) \text{ is the Lebesgue measure set}\)
\( E \text{ is a null set on } ( \Sigma , X , \mu ) \iff \forall F \subseteq E, \mu (E) = 0\)
\( E \text{ is positive on } ( \Sigma , X , \mu ) \iff \forall F \subseteq E, \mu (E) \geq 0 \)
\( E \text{ is negative on } ( \Sigma , X , \mu ) \iff \forall F \subseteq E, \mu (E) \leq 0 \)
Measure space where all null sets are in the Sigma algebra
\( (\Sigma, X, \mu) \text{ is a complete measure space } \iff \forall A : \mu(A)=0, A \in \Sigma \)
Sequence space defined by the following:
\(\ell^p = \{ x_n : (\sum^{\infty}_{i=1} |x_i|^{p} )^{\frac{1}{p}} \lt \infty \}\)
Quasi-norm (function similar to a norm but not exactly) defined on \(L^p\) spaces as such:
\(\| x \|_{p} = (\sum_{i=1} |x_i|^p )^{\frac{1}{p}}\)
\(\| f \|_{\infty} = \sup |f(x)| \)
Function space defined by the following:
\(L^p = \{ f : (\int |f|^p d \lambda)^{\frac{1}{p}} \lt \infty \}\)
\(L^{\infty} = \{ f : f \text{ is bounded a.e.} \}\)
Quasi-norm (function similar to a norm but not exactly) defined on \(L^p\) spaces as such:
\(\| f \|_{p} = (\int |f|^p d \lambda )^{\frac{1}{p}}\)
\(\| f \|_{\infty} = \text{ess sup} |f(x)| \)
\( \int fg d \lambda \leq |f|_{p} |g|_{p} \)
\( L^{p} \text{ is almost a Banach space} \)
\( L^{2} \text{ is almost a Hilbert space} \)
\( f \in L^p \land g \in L^q \land \frac{1}{p} + \frac{1}{q} = 1 \implies\)
Triangle inequality in \(L^{P}\) spaces
\( f,g \in L^p \land 1 \leq p \leq \infty \implies f+g \in L^p \land \|f+g\|_{p} \leq \|f\|_p + \|g\|_{p}\)
\( (X, \Sigma, \mu) \text{ is a sigma-finite signed measure space } \land \nu \text{ is a measure on } \Sigma \implies \exists! \nu_0 , \nu_1 : \)
Consider a series of measurable spaces \( { (X_i , \Sigma_i , \mu_i ) }^{\infty}_{i=1}\), \( (X,\Sigma, \mu) \)is an infinite product space
3-tuple \( (\Omega, \mathcal{F}, \text{Pr}) \)
Fundamental laws of probability (adjusted from a branch of mathematics called measure theory', see Lebesgue Integration and Fourier Analysis)
Analogue of "almost everywhere" in the sense of a probability space
\(E \text{ occurs almost surely } \iff \text{Pr} (E \setminus N) = 1 \land \text{Pr} (N) = 0\)
\(\sigma\)-finite measure space that obeys the Kolmogorov axioms
3-tuple \( (\Omega, \mathcal{F}, \text{Pr}) \) such that:
Analogue of "almost everywhere" in the sense of a probability space
\(E \text{ occurs almost surely } \iff \text{Pr} (E \setminus N) = 1 \land \text{Pr} (N) = 0\)
On a probability space \( (\Omega, \mathcal{F}, \text{Pr}) \), an SRV is a \(\mathcal{F}\)-Measurable function that maps all outcomes in the sample space to a result in the state space, a state space being a numerical representation for an outcome.
\(X : ( \Omega , \mathcal{F} ) \to ( R , \mathcal{G} ) \text{ is a random variable in } ( \Omega , \mathcal{F}, \text{Pr}) iff X \text{ is measurable} \)
\( \text{E}[g(\textbf{x})] = \int_{\Omega} g[X(\omega)] d\text{Pr}(\omega ) \)
\(\text{E}( \textbf{x} ) = \begin{bmatrix} \text{E} (X_1) \\ \text{E}(X_2) \\ \vdots \\ \text{E}(X_n) \end{bmatrix} \)
\( X \in L^1 (\Omega) \implies \text{E}[g(X)] = \int_{\Omega} g[X(\omega)] d\text{Pr}(\omega ) \)
\( X \in L^1 (\Omega) \implies \text{E}(X) = \int_{\Omega} X(\omega) d\text{Pr}(\omega)\)
\( X \in L^1 (\Omega) \implies \text{E}(X) = \int_{\Omega} X d\text{Pr}\)
\( X \in L^2 (\Omega) \implies \text{Var}(X) = \text{E}([ X -\text{E}(X)]^2)\)
\(\text{Pr}(A|B)= \frac{\text{Pr}(A \cap B)}{\text{Pr}(B)}\) is a measurable function defined almost everywhere (consider when \(B\) is a null set).
For a sigma algebra \(\mathcal{F}\), consider a sub sigma algebra \(\mathcal{B}\) and denote its sets as \(B \in \mathcal{B}\), then \(\text{Pr}(A \| \mathcal{B} ) \) is t
Consider a measure \( \mu(B)= \text{Pr}(A \cap B), B \in \mathcal{B} \), since \(\mu \ll \text{Pr}\) (All null sets of \(\text{Pr}(B)\) are null sets of \(\text{Pr}(A \cap B)\)), RNT implies \(\text{Pr}(A\cap B) = \mu(B) = \int_B f d \text{Pr}\) where \(f = \text{Pr}(A|B) \)
\(E_{\text{Pr}}(f \| \mathcal{B}) \)
Stochastic process in a Banach space \(X_n\) such that:
Ordered pair of a sequence of random variables and sigma algebra \( (f_n, \mathcal{A}_n ) \) such that:
These conditions characterise random variable sequences such that the expectation on the previous sigma algebra is given by the previous random variable;
Theorem asserting the convergence of martingale random variable sequences
\((f_n ,\mathcal{A}_n ) \text{is submatringale} \implies \exists f :\)
\((f_n \to f \text{ pointwise a.e.}\)
\(( \displaystyle \int |f| d\text{Pr} \leq \sup_n \int |f_n| d\text{Pr}\)
\( E_{\mu} (f \| E)^p \leq E_{\mu} ( |f|^p \| E ) \)
\((f_n, \mathcal{A}_n ) \text{ is submartingale } \land \alpha \gt 0 \implies \text{Pr} \{ \max_{i \leq n} f_i \geq \alpha \} \leq \frac{1}{\alpha} \int |f_n| d \text{Pr}\)
Consider \( [a,b] \), the amount of upcrossings is the amount of times a sequence passes from below \(a\) to above \(b\)
\( (V, \| \cdot \| ) \text{ is a Banach space } \iff \forall S_n \in (V, \| \cdot \| ) , (S_n \text{ is absolutely convergent } \implies S_n \text{ is convergent} ) \)
\( (H, \langle \cdot \rangle ) \text{ is a Hilbert space } \land K \subset H \text{ is an orthonormal set} \implies \)
Binary function returning a scalar related to the orthogonality of the two linear elements
Unary function returning the magnitude of an element in a linear space
\( \| \textbf{v} \| \geq 0\)
\( \| \textbf{v} \| = 0 \iff \textbf{v} =\textbf{0} \)
\( \| \textbf{v} + \textbf{u} \| \leq \| \textbf{v} \| + \| \textbf{u} \| \)
\( \| c\textbf{v} \| = |c| \| \textbf{v} \|\)
Property generalizing perpendicularity to two elements in a linear space
\( \textbf{v} \perp \textbf{u} \iff \langle \textbf{v} , \textbf{u} \rangle = 0 \)
\( | \langle \textbf{u} , \textbf{v} \rangle | \leq \| \textbf{u} \| \| \textbf{v} \|\)
\(L^p \text{ is a Banach space}\)
Orthonormal basis of \(L^2\)
\(\text{He}_n(x) = (-1)^n e^{\frac{x^2}{2}} \frac{d^n}{dx^n}e^{-\frac{x^2}{2}}\)
\(H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}e^{-x^2} \)
Integral transform \(\mathcal{F}\) on functions in \(L^1 (\mathbb{R})\) that describes how periodic functions of frequency \( 2 \pi \xi \) are present in a function.
\( \displaystyle \mathcal{F} \{ f \} (\xi) = \widehat{f}(\xi ) = \int^{\infty}_{-\infty} f(x) e^{-i 2 \pi \xi x} dx \)
\( f,g \in L^1 (\mathbb{R}) \)
Lemma asserting that the any Fourier transform on an \(L^1(\mathbb{R}^n)\) function tends to 0
\(f \in L^1 (\mathbb{R}^n) \implies \lim_{\xi \to \pm\infty} \mathcal{F}\{f\}(\xi) = 0\)
This result also applies to Fouier series coefficients
\(f \text{ is continuous on } I \land f \text{ is } 2\pi \text{periodic on } \mathbb{R} \implies \lim_{n\to\infty} |\widehat{f}(n)|=0\)
Operation on \(L^1(\mathbb{R})\) functions such that the integral of \(f\) is evaluated with \(g\) as a weight function that is translated by \(t\)
\( (f * g)(t) = \int_{-\infty}^{\infty} f(\tau )g(t -\tau) d \tau \)
\( (f*g) = (g*f) \)
\( \| f * g \|_1 \leq \| f \|_1 \| g \|_1 \)
\( f,g \in L^1 (\mathbb{R}) \implies \mathcal{F}\{f * g\}(\xi) = \mathcal{F}\{f\}(\xi)\mathcal{F}\{g\}(\xi) \)
\( \displaystyle \mathcal{F}^{-1} \{f \} (x) = \int^{\infty}_{-\infty} f( \xi ) e^{i 2 \pi \xi x} d\xi \)
\(f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R}) \implies \| \mathcal{F}\{f\} \|_2 = \| f \|_2\)
\( \displaystyle \mathcal{F}\{f\}(\boldsymbol{\xi}) = \int_{\mathbb{R}^n} f(\textbf{x}) e^{-i \langle \textbf{x} , \boldsymbol{\xi} \rangle } d\textbf{x}\)
Integral transform that extends the notion of Fourier transform for radial functions
\( \displaystyle \text{H}_n\{f\}(\rho ) = \int_{0}^{\infty} f(r) J_{n}(\rho r)r dr \)
Group of the complex numbers of magnitude 1, on which fourier transofmr shvae interesting proverties
\( \textbf{T} = \{ z \in \mathbb{C} : |z| = 1 \} \)
Space \(L^{1}( (-\pi, \pi] )\). The following hold:
Infinite series on a periodic space \( L^1( (-\pi , \pi] ) \) relating to a periodic function \(f\)
Some functions can be proved to equal their own Fourier series
\(\displaystyle f \sim \sum^{\infty}_{n = -\infty} \widehat{f}_n e^{inx}\)
\(\displaystyle f \sim a_0 + \sum^{\infty}_{n = 1} a_n \cos (n x ) + b_n \sin (n x) \)
For periodic space \( L^1( (-L , L] ) \) relating to a periodic function \(f\)
\(\displaystyle f \sim a_0 + \sum^{\infty}_{n = 1} a_n \cos (\frac{n \pi x}{L} ) + b_n \sin (\frac{n \pi x}{L}) \)
\(\displaystyle f \sim \sum^{\infty}_{n = -\infty} c_n e^{i\frac{n \pi x}{L}} \)
Theorem relating integral of a function to a series involving its Fourier coefficients
\( f \in L^2 \implies \frac{1}{2\pi} \int^{\pi}_{-\pi} |f (x)|^2 dx = \sum^{\infty}_{n=-\infty} | \widehat{f}(n)|^2 \)
\( f \in L^2 \implies \frac{1}{\pi} \int^{\pi}_{-\pi} |f (x)|^2 dx = 2a^2_0 + \sum^{\infty}_{n=1} a^{2}_{n} + b^{2}_{n} \)
\( D_n (x) = \sum^{n}_{k=-n} e^{ikx} = \frac{ \sin ( \frac{n+1}{2} x )}{ \sin ( \frac{x}{2} ) }\)
\( F_n(x) = \frac{1}{n} \sum^{n-1}_{k=0} D_k (x) = \frac{ \sin^2 ( \frac{n}{2} x )}{ n \sin^2 ( \frac{x}{2} ) }\)
\( F_n(x) = \frac{1}{n} \sum^{n-1}_{k=0} D_k (x) \)
\( \forall x, F_n(x) \geq 0 \)
\( \int^{\pi}_{-\pi} F_n(x)dx = 2\pi\)
\(\forall \delta \in (0,\pi) \lim_{n \to \infty}\int_{\delta \lt |x| \lt \pi} F_n(x)dx = 0\)
\( \lim_{n \to \infty} T_n(x) \to S_n(x) \text{Uniformly}\)