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For the purpose of simplicity, assume everything below is only in the discrete domain.

A $\text{probability space}$ is usually defined as a triple $(\Omega , 2^\Omega , P)$ where

$\Omega := \text{set of outcomes}$

$2^\Omega:= \text{set of events (for simplicity)}$

$P:= \text{mapping from } 2^\Omega \mapsto [0,1] \text{ which assigns a probability to each event}$

My first question is whether $P$ is a probability distribution or a probability measure, and what exactly is the difference between these two ideas.

Moreover, I am trying to understand where the concept of a $\text{random variable}$ fits in with all this.

From my reading a random variable (over the probability space above) $X:= \text{mapping from } \Omega \mapsto \mathbb{R}$.

When we say $P(X=a)$, is this $P$ referring to the same probability distribution of the space? What exactly is the object $X=a$? I would think it refers to the set of all outcomes in $\Omega$ that map to $a$ but my reading does not elucidate this.

What is the ultimate confusion is the statement: $\text{we will use the notation} P(X) \text{ to denote the distribution of the random variable X}$

Again, what exactly is meant by the probability distribution $P$ here? Does it also refer to the $P$ of the probability space that $X$ is defined over? How is this distribution different?

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Probability distribution and probability measure are synonyms.

$[X=a]=X^{-1}(\{a\})=\{\omega\in\Omega\mid X(\omega)=a\}$ hence $P(X=a)=P(X^{-1}(\{a\}))$.

The distribution of the random variable $X:\Omega\to\mathbb R$ is the unique probability measure $\mu$ on $\mathcal B(\mathbb R)$ defined by $\mu(B)=P(X\in B)$ for every $B$ in $\mathcal B(\mathbb R)$, where $[X\in B]=X^{-1}(B)=\{\omega\in\Omega\mid X(\omega)\in B\}$.

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    $\begingroup$ great response, but what does B(R) mean? $\endgroup$ – jII Apr 30 '13 at 22:49
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    $\begingroup$ This is a terrible answer in my opinion, no matter OP accepting it. The question is clear, concise and directed towards beginners of the topic. You, on the other hand, answer in an obscure and expert manner. Someone starting to study probability does not know what a borel sigma-algebra is, and surely will just get confused on your answer. If you introduce a new concept, you should define it, and show how this fits into your answer. $\endgroup$ – Euler_Salter Dec 1 '17 at 9:40
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    $\begingroup$ For instance, how does a beginner know that $X$ admits an inverse $X^{-1}$? Does even $X^{-1}$ refers to the inverse? What is the meaning of the notation $[X=a]$ and why is it connected to $X^{-1}$. And moreover what is the difference between the probability measure and the random variable? That was the question, which you didn 't answer. Furthermore, saying WHAT something is, (yes also in mathematics!) is not equivalent to saying WHY something is what it is. Yours are definitions, which are absolutely meaningless to a beginner. $\endgroup$ – Euler_Salter Dec 1 '17 at 9:43
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    $\begingroup$ @Did it depends on the definition of he Borel sigma algebra, which again is a new concept introduced by you. So for someone who doesn't know that, $X^{-1}$ is still rather too abstract and undefined. $\endgroup$ – Euler_Salter Dec 1 '17 at 13:20
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    $\begingroup$ @Did you use the symbol $=$ for both definition and equality symbol, these are not the same as the former you assume, hope or simply say that is so, in the latter case you are necessarily using some theorem, axiom, or some way of reasoning that in the end you need to be able to prove. Thus if you write $[X=a]=X^{−1}(\{a\})=\{\omega\in\Omega∣X(ω)=a\}$, in my standard math world this is equivalence, not definition, thus the reader is supposed to be able to prove such equivalence. (and I appologize for missing the curly braces in my comment). $\endgroup$ – leosenko Dec 29 '17 at 13:28

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