Theory where Probability zero = Impossible event? Is it considered a paradox that in probability theory, possible events can be given a probability of $0$? For example, if we pick out a random person and measure their height, we give it probability zero that the height will be exactly $170$ centimeters ... but, assuming we could measure heights as precisely as we want, it's still possible that such a person is drawn out of the sample group.
Do there exists an axiomatic setting where this isn't the case, i.e. where one can avoid this type of 'paradoxes'? (I put it in quotes, because I don't think it's controversial at all, but am wondering if maybe other people think it is).
 A: How do we measure height?
We place a person against a wall, we take a ruler and count the number of ticks on the ruler until we reach his height. Normally, there is some uncertainty, because the person is breathing, and the ruler only has a certain density of ticks, so generally we can take an estimate and say that person Yanni is between $185.5$cm and $186.0$cm.
Say we devise a more elaborate way of measuring height, perhaps with lasers or whatnot. There will still be some uncertainty: Yanni is growing and his atoms are vibrating etc. But now we can say that Yanni measures between $185.567$cm and $185.583$cm.  
Keep this in mind while I explain the framework in which we deal with it.
Probability density
If we assume that each individual's height is a real number, and that there is an infinite amount of individuals, we can describe the distribution of individual heights in the following way.
There is a function $f$ called the probability density function which contains all the information. It has the following properties:


*

*For any height $x$, $f(x)$ is a non-negative number.   

*The integral $\int_0^\infty f(x)~dx=1$.


Now, $f$ can tell us something very important. Take a random individual, and let $y_1$ and $y_2$ be two heights. Then the probability that this random individual's height is between $y_1$ and $y_2$ is given by the integral:
\begin{equation}
\int_{y_1}^{y_2}f(x)~dx
\end{equation}
Back to the original problem
So now you see, in this framework, the number $y_2-y_1$ corresponds to the accuracy of your measurements. As you make it smaller, the probability decreases as you are guessing the height to a smaller and smaller precision.
You see, that the probability of Yanni's height being exactly $y$ is $\int_{y}^{y}f(x)~dx = 0$. This is true no matter what $y$ you choose! "But Yanni must have a height", you might say. And this is where things get more subtle. As we said, Yanni's height can be measured, but always with uncertainty. Both because of the precision of your apparatus, and the fact that Yanni's height actually constantly changes! So one could be happy with knowing that Yanni's height is between certain two numbers.   
The `paradox'
In general, one is then interested in the probability of measuring a certain value to lie in a specific range, rather than taking a specific value. This is because the measuring is always with some uncertainty. So one does not bother that the mathematical model we use, the probability density framework, assigns probability only to ranges and not specific numbers.
