Measure theory in the rationals.

Abstract

For the graduate math student with an interest though little background in measure theory, Measure Theory in the Rationals presents a simplified look at the theory's development. The idea, just like measure theory in the reals, is to develop complex measurement ideas using familiar, simple objects. These objects include rational numbers and lengths of intervals of rational numbers, We start with an agreement that we will work only with rational numbers as if irrationals do not exist From here we seek a function that matches any set of rationals, not just intervals, to a unique number that describes the set's "size". This "size" is called the set's quasi-measure. This sought-after function should have special properties and we set out to find a function which best obtains the ideal properties we have in mind. It's all a matter of give and take as the most ideal properties prove to be impossible to attain at once. What is achieved is summarized and then compared to its parallel in the real numbers. Not only does the reader see similarities in measure theory's development between the rationals and the reals, but sees the contrast between the rationals and reals themselves.