Existence of a measure on semi-algebra

Question

Suppose a semi-algebra (/algebra/sigma algebra) $\mathbb{S}$ is given. I know the definition of measure $\mu$ on $\mathbb{S}$. But how can I show the existence of such measure? If we have a measure on Semi-algebra then we can extend it to sigma algebra. But to extend the measure I have to have a measure on the Semi-algebra at the first place. Right?

Answer 1

Defining the measure on the semi-algebra will be done differently for different examples. For example, in the real line, your semi-algebra is all intervals of the form $[a,b)$ and the measure of $[a,b)$ is $b-a$.