Question on Integration

I have been leading chapter 3: Stochastic Integration in Edward Allen’s book on Modeling with Ito Stochastic Differential Equations, he has similar approach of Stochastic Integration as Riemann’s idea, start with step function, simple function, then using sequences of simple functions to approach functions to form integration. 

I wonder if there is a version of Lebegues integration for stochastic? Or at least is it legitimate to do this?

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