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If I want to prove a result regarding $\int f$ where we are considering the Lebesgue integral in this case, it would be best to investigate what happens for $\int \chi_E$ where $\chi$ is a simple function with $E$ being a set of finite measure. With this at hand we can use the fact that for any measurable $f$ there exists a sequence of simple functions which are increasing in the absolute value sense. Consequently we can then apply the monotone convergence theorem to achieve the result for $\int f$.

But now, it seems that it is easier to investigate what happens when step functions are used instead of simple functions. My question now is that how can I express a simple function in terms of step functions in $\mathbb{R}$? And how will I relate $\int \chi_I$ where $I$ is an interval and hence $\chi_I$ is a step function with $\int \chi_E$ where $\chi_E$ is the simple function?

Help appreciated!

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