 The underlying probability space formula_4 is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space.In practice, one often disposes of the space formula_4 altogether and just puts a measure on formula_49 that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.See the article on quantile functions for fuller development.