I've always heard and read the sentence:

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> If you pick a real number $x\in[0,1]$ **at random**, the **probability** to obtain a rational number is $0$.
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**What is the meaning for that? Is this the "real" probability?**

First, we need to define accurately what means *at random*, I'll assume a base $10$. Then all numbers in $[0,1]$ begins with $0$ (for $1$ we discard this notation and we will use $0.9999\ldots$).

Then I'll give a classical approach defining random meaning, if we pick a real number $x$ in $[0,1]$ *at random* we mean that the probability to get a digit in certain position is $1/10$ (extending for $[0,1]$ the Laplace classical definition for probability), roughly speaking each digit has the same "probability" to be in $x$.

Strictly speaking we are building a notion of probability in $\{0,1,2,3,4,5,6,7,8,9\}^\omega$ since numbers with a $0$ periodic has two representations. We want to pass the intuitive notion of probability to this set, but if there is one, this must be unique.

The Lebesgue measure $m$ rescues some of these probabilistic notions for example $m(\emptyset)=0$, $m(\{x\})=0$, and $m(\text{numbers starting with the given digit $k$})=\frac{1}{10}$, but intuitively a probability notion must be defined in all subsets (**Edit:** sos440 in comments says that is natural this phenomenon (not defined in all subsets) since we can't describe some sets without using additional axioms like "**Zorn Lemma**") apart of having some properties like (finite or $\sigma$)-additivity.

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> Lebesgue measure also satisfies $m(\mathbb{Q\cap [0,1]})=0$. **Does our intuitive notion of probability tell us that the probability to get a rational number is $0$?**
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Other functions("measures") in $[0,1]$ are defined in whole $\mathcal{P}([0,1])$ but doesn't coincides with $m$ and we loss other notions for our intuitive probability meaning.

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> **Does Lebesgue measure deserve to be called a "real" probability in $[0,1]$ (or $\{0,1,2,3,4,5,6,7,8,9\}^\omega$)? Why?**
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> This was completely answered by Michael Greinecker ♦ The answer is a big *Yes*.
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> **Is possible to define a probability notion for $[0,1]$ and for whole $\mathbb{R}$?**
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> Fully answered in comments and answers. The answer is no for $\mathbb{R}$, and for $[0,1]$ is yes.
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> **Is there any function(measure) that is the best representative for our concept of probability? Why?**
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> This was completely answered by Michael Greinecker ♦: The Lebesgue measure.
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Thanks in advance.

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> **Second Edit:** People in comments and answers talk me about importance of additional axioms like "Axiom of Choice" and "Continuum Hypothesis". How acceptance or denial of these axioms affects the answers to my questions here?
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> This was fully answered by hot\_queen.
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**Third Edit:** I marked the questions were completely answered and offered a bounty for intuitive answer and essencially the first question about $\mathbb{Q}$.