Uncountable infinity The "number" of real numbers in $[0,1]$ is uncountably infinite, just as the "number" of real numbers in $[0,10]$ is uncountably infinite. However, my intuition would tell me the second interval has many more real numbers in it than the first interval. And what about the "number" of real numbers in $(-\infty, \infty)$? Are there the "same number" of real numbers in all those intervals?
 A: The function $f:[0,1]\to[0,2]$, where $f(x)=2x$ it is a bijection. That bijection tells us that $[0,1]$ has as many elements as $[0,2]$.
But 
$$\int_{[0,2]}d\mu=2\int_{[0,1]}d\mu$$
tells us that the measure of $[0,2]$ is twice bigger than that of $[0,1]$.
A: Even though [0,10] intuitively has a longer length than [0,1], the "amount of numbers" in both intervals is the same, i.e. the sets have the same cardinality.  To see if two sets have the same cardinality, you must create a bijection between them, i.e. a map that is both surjective and bijection.
For example, consider the function $f:[0,1]->[0,10]$ such that $f(x)=10x$.  This map would serve as your bijection, which in turn means that your domain and codomain have the same cardinality.
As for your second question, indeed the set of real numbers is again uncountable, and hence "have the same amount of points" as your interval [0,1].
A similar idea that works for countably infinite sets would be to show that the set of natural numbers (1,2,3,4,5,...) has the same cardinality of all the even numbers, even though there are "twice as many" natural numbers than there are even.
A: We say that two sets have "the same number of elements" (it is better to say that they have the same cardinality), iff there exists a bijection between these two sets.
In the case of  $[0,1]$ and $[0,10]$ this bijection is trivial: $x\to 10 x$. In the case $[0,1]$ and $(-\infty,\infty)$ is a little trickier: first, we show that $[0,1]$ has the same cardinality as $(-1,1)$ and then build an explicit bijection $x\to \tan (\pi  x/2)$ to $(-\infty,\infty)$.
Recall also the definition of infinite sets: a set infinite iff it has the same cardinality as its non-trivial subset.
