If we define a measurable function on a probability space as a function that maps from the sample space to the real numbers, and treat all the people in a Country as the sample space, shouldn't the image of the random variable be a finite set? I don't understand why we would say that the random variable is countable.
Assuming data are normal is a practical convenience. A discrete random variable that has very many possible values requires a long list of possible values and associated probabilities.
In the US, heights of people are usually rounded to the nearest inch. (A person's height can fluctuate by about half an inch during the day--usually taller in the AM shorter in the PM. So rounding to the nearest inch doesn't lose crucial information.)
Almost all US adults are between 50 and 90 inches tall, so in using a discrete random variable there would be about 40 different values to list along with probabilities of each. Saying that heights are approximately normally distributed with mean 68 and standard deviation 3.5 allows us to get by with 2 parameters.
Theoretically, a normal distribution takes all values on the real line, but an interval $(56,89)$ with $99.9\%$ of the probability is manageably short--even if it theoretically contains uncountably many values. Also one ignores the fact that this normal distribution allows a very tiny probability of (practically impossible) negative heights.
According to this normal distribution, what is the probability that a randomly chosen person is 68in tall? The theoretical answer is $0.$ (if you have uncountably many points you can't assign positive probabilities to any of them.) If you mean $68.00000$ inches tall, then the practical answer is also essentially $0.$ (Who could measure that?) It is more useful to ask for the probability that person is between $67.5$ and $68.5$ (so that the height would be rounded to "68". Then the answer is $0.1136.$
If pilots for a small military airplane need to be shorter than 65 inches, then almost 20% of the population (including lots of women) are still potential candidates as pilots. All such computations are understood to be approximate. Very few, if any, statisticians believe any natural phenomenon is exactly normally distributed.
Sometimes, the gulf between measure theoretic probability and applied statistics can be very wide.
Computations using R software, where
a continuous normal cumulative distribution function, and
qnorm is its inverse (called
a quantile function),
qnorm(c(.0005,.9995), 68, 3.5)  56.48316 79.51684 pnorm(0, 68, 3.5)  2.212535e-84 diff( pnorm(c(67.5,68.5), 68, 3.5) )  0.113597 pnorm(65, 68, 3.5)  0.195683