I am self studying Apostol Mathematical Analysis Ch-> Lebesgue integration and I have a different thinking about a proof whose only outline is given in Apostol Book.
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In part (a) when Apostol writes $g \in L(I+c)$ and in proof Apostol gives First verify theorem for step functions, then upper functions and then Lebesgue-integrable functions.
But I thought as $f \in L(I)$ and $g(x) = f(x-c)$ for $x \in I+c$. So,using it $g \in L(I+c)$ as $f \in I$ can be written as difference of upper functions and using the note I can derive the integral relations in all 3 cases (which I have done in note book).
But I never used any of verification on step functions, then for upper functions and finally Lebesgue-integrable functions.
So, am I right in my approach? and if not kindly tell the right way to prove it.