Picking a positive real number The question I have is rather "simple" in its statement but I need to go deep in its meaning and solutions, if it makes sense.

Let's take the set $\mathbb{R}$ of the real numbers. What is the probability of picking a positive real number? 

If one reasons in a very naif way, one would say $1/2$ for you might pick a positive real or a negative real number. 
But it's also true that there is no probability to the positive real numbers that assigns equal weight to every real. Am I right? 
So one should work with probability densities? 
Or are there clearer methods to analyse this problem? 
 A: Your displayed statement poses an incomplete question.  The answer depends on what "probability" means in this context. There are, for each $p\in[0,1]$, many probability laws making "$p$" the correct answer.   
Perhaps you also believe that the unstated probability distribution is symmetric about $0$?  (Then mass at the origin makes any answer $p\in[0,1/2]$ valid, for some probability law.)  Or some other structural property of the unstated probability law.  And so on.
You are right, there is no  translation-invariant probability law on $\mathbb R$.
A: 
"What is the probability of picking a positive real number?"

That is still to be "decided" and cannot be answered on forehand.
What we need (and there a decision comes in) is a probability space $(\mathbb R,\mathcal A,P)$ that satisfies $(0,\infty)\in\mathcal A$.
That makes it possible to say something about the probability of picking a positive real number: $$\text{this probability equals: }P((0,\infty))$$If you like you can take a probability measure $P$ that satisfies $P((0,\infty))=0.5$, but that is not a necessity.
There are lots of choices and if we want to say something sensible then we are forced to make one. 
In the absence of such a space nothing can be said about the probability of picking a positive real number.
