
> 
> **Possible Duplicate:**  
> 
> [Why is Lebesgue integration taught using positive and negative parts of functions?](https://mathoverflow.net/questions/25161/why-is-lebesgue-integration-taught-using-positive-and-negative-parts-of-functions) 
> 
> 
> 


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!



Why are simple functions defined for positive coefficients (in measure theory) 
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.


EDIT: I didn't see that Gerald's comment already links to that answer, sorry for the duplicate.


