Consider the set of all points on the circumference of a circle and the set of all points inside a circle. what is difference in the measure of these two sets.
The circumference of a circle has a measure of length (e.g. inches) and the set of all points inside the circle has a measure of area (e.g. square inches).
Cantor proved that there are not "more" (as measured by one-to-one correspondence) points in the area of a circle than there are on the circumference! So it is wrong to thing of measures simply a count of the number of points.
Is Bertrand paradox related to measure theory?
Absolutely. That is the reason that the paradox is so popular. The paradox twists three measures together in what is confusing until you apply measure theory.
Can Bertrand paradox be resolved with the notion of a measure of a set.
Essentially, but it is more in understanding how the different ways that different measures and different sets can interact.
If so please suggest some good references/books for measure theory.