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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!