The primary focus of my PhD did not turn out as theoretical as I would have hoped, and I never found time to dig very far into measure theory books. Keeping notes of the concepts I read about in this blog post will hopefully help me dedicate time to this. It might also help some interested readers in their journey through measure theory !

Real analysis

lim sup of sum of arbitrary real sequences

Let \((x_n)_{n\in\mathbb{N}}\) and \((y_n)_{n\in\mathbb{N}}\) two series in \([-\infty, +\infty]\). Then,

\[ \begin{aligned} &\forall\, n\in\mathbb{N},\,\forall t>n, \quad x_t \leq \sup_{t>n}(x_n) \, \text{and}\, y_t \leq \sup_{t>n}(y_t).\\ \text{Therefore } &\forall n\in\mathbb{N},\,\forall\, t>n, \quad x_t + y_t \leq \sup_{t>n}(x_n) + \sup_{t>n}(y_t)\\ \text{therefore } &\forall n\in\mathbb{N},\,\forall t>n, \quad x_t + y_t \leq \inf_n\sup_{t>n}(x_n) + \inf_n\sup_{t>n}(y_t)\\ \text{therefore } &\forall\, n\in\mathbb{N}, \quad \sup_{t>n}(x_t + y_t) \leq \inf_n\sup_{t>n}(x_n) + \inf_n\sup_{t>n}(y_t)\\ \text{therefore } &\inf_n\sup_{t>n}(x_t + y_t) \leq \inf_n\sup_{t>n}(x_n) + \inf_n\sup_{t>n}(y_t)\\ \text{i.e. } &\lim\sup (x_n + y_n) \leq \lim\sup (x_n) + \lim\sup (y_n) \end{aligned} \]

lim sup of sum of one arbitrary and one converging real sequences

Suppose that \(\lim y_n = a<\infty\). \(\lim\sup(x_n)\) can be understood as the supremum of all the limits of converging sub-sequences of \((x_n)_{n\in\mathbb{N}}\). Let \(L\) denote the set of the limits of the converging sub-sequences of \(x_n\).

Then, the set of the limits of all the converging sub-sequences of \(x_n + y_n\) is \begin{equation} L+a={t+a,|,t\in L}.\end{equation}

Then we have:

\begin{equation} \lim\sup (x_n + y_n) = \sup (L+a) = \sup (L) + a = \lim\sup(x_n) + \lim y_n \end{equation}

Measure Theory

Probability density function of a random variable

Let \(K=\mathbb{R}\) or \(\mathbb{C}\) (assume \(K=\mathbb{R}\)). ? K could be any field (corps commutatif)?

Let \(E\subseteq K\). Let \((E,\,\mathcal{B},\,\lambda)\) the Lebesgue (Borel) measure space.

Let \(\Omega\) a set. Let \(\Sigma\) be a \(\sigma\)-algebra on \(\Omega\). Let \(\mathbb{P}:\,\Sigma\,\rightarrow\,[0,1]\) be a measurable function such that \(\mathbb{P}(\Omega)=1\) (+ some other assumptions that I skipped).

Let \((\Omega,\,\Sigma,\,\mathbb{P})\) be a probability space.

Continuous Random Variable

Let \(X:\,\Omega\,\rightarrow\,E\) be a measurable function. The measurable functions defined in a probability space are called random variables.

Probability Distribution

The probability distribution of the random variable \(X\) is the measurable function:

\begin{equation}p_{X}:,\Sigma,\rightarrow,[0,1], \end{equation}

defined wrt to \(\mathbb{P}\), that assigns a probability to every possible element of the \(\sigma\)-algebra such that:

\begin{equation} \forall S,\in,\mathcal{B},,p_{X}(S)=\mathbb{P}(X \in S)=\mathbb{P}({\omega \in \Omega,|,X( \omega ) \in S}). \end{equation}

The probability distribution of the random variable \(X\) is a measure on \(E\), that was “pushed-forward” by \(X\), from the measure \(\mathbb{P}\) on \(\Omega\).

Density Function

A density function of \(X\) is the Radon-Nykodym derivative of \(p_X\) wrt to \(\lambda\). Let us assume that

\begin{equation} \forall S,\in,\mathcal{B},,\lambda (S)=0,\implies,p_{X}(S)=0, \end{equation} i.e. \(p_{X}\) is absolutely continuous wrt \(\lambda\), also noted \(p_{X} << \lambda\).

Then we have that: \(\exists ! \lambda-a.e\, f:\Omega\rightarrow [0,+\infty[\) such that,

\begin{equation} \forall A\subseteq \Omega,,p_{X}(A)=\int_{A} f d\mu \end{equation}

Discrete Random Variables

Let us now assume that the initial measure space is defined with (1) a finite \(\sigma\)-algebra \(\mathcal{B}\) (i.e. containing singleton sets of \(B\) and their unions) and (2) the counting measure \(\mu:S\mapsto |S|\) on \(\mathcal{B}\). A random variable \(X:\Omega\rightarrow E\) is discrete iif \(E\) is countable. if X is a discrete random variable the probability distribution of X is the push forward measure \(f_X\) defined as:

\begin{equation} \forall x\in V f_X(x)=\mathbb{P}(X=x) = \mathbb{P}({\omega\in\Omega|X(\omega)=x}). \end{equation}

Supposing that the Radon-Nykodym derivative of \(p_X\) wrt to the counting measure exists