# What is the usual definition of measure?

This is my first time trying to learn measure and i found there are several different definitions for measure which are not equivalent.(i.e. $\sigma$-ring, $\sigma$ field, $\sigma$ algebra etc)

I want to know not only the usual definition for measure but also differences between those definitions too! (i.e. Universal set)

Measures are defined on $\sigma$-algebras. The notion of a $\sigma$-ring is useful for developing measure theory but it is not what measures are usually defined on. – Qiaochu Yuan Jan 4 at 21:39
@Qiaochu Rudin PMA defined measure on $\sigma$ ring, so is it better to skip this chapter and study another book? If so, would you please recommend me a book which briefly but precisely defines measure? I just want to define measure, Lebesgue Integral and go back to Fourier Analysis – Katlus Jan 4 at 21:46
Also note that $\sigma$ algebra and $\sigma$ field are typically used interchangeably, even though (obviously) the naming algebraic structures are not the same. – gnometorule Jan 4 at 23:18