# Why are simple functions defined for positive coefficients (in measure theory) [duplicate]

Hey,

I am currently referring 'probability with martingales'. To develop lesbegue integration they have first defined it over simple functions.

where a simple function is defined as sigma { (a_i) * I_Ai }

where each a_i is positive, and Ai form a partition over the sample space. I_Ai is the indicator function taking value for elements in A_i and 0 else.

My question being why is the condition of positivity on a_i needed? To the best of my knowledge it isn't used in derivation f its properties.

Further then integral is defined separately for positive functions and then for general functions (writing it as difference of two positive functions)

My question is, why such different approach to it? Why not define it directly for general functions?

Thanks!

-

## marked as duplicate by S. Carnahan♦Mar 19 '12 at 7:09

The positivity is absolutely not needed. Serge Lang's Real Analysis directly develops Lebesgue integration of Banach-space-valued functions, so his simple functions are not positive. Not only is it much cleaner than the $f_+-f_-$ business, it is also more general since the "standard" approach can only be extended to functions taking values in finite-dimensional vector spaces.