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While self studying Lebesgue Integration from Apostol Mathematical Analysis I am struck on this theorem

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I am not able to understand how if theorem is true for {$s_n$ - $s_1 $} then it's true for $ s_n$ also.

I tried by putting { $s_n$ -$s_1$} everywhere in theorem where I have {$ s_n$ } but if $\int_{I} f $ = lim n->$\infty$ $\int_{I} ( s_n -s_1)$ then how does integral of $s_n$ also converges to same limit of f ie $\int_{I} f $?

I am confused here and can anyone please derive the required justification.

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1 Answer 1

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You get that $\{s_n-s_1\}$ converges almost everywhere on $I$ to a limit function $f$ in $U(I)$, and $$ \int_If=\lim_{n\rightarrow\infty}\int_I(s_n-s_1). $$

Your mistake was to try to prove that $\{s_n\}$ converges to $f$. But that's not needed. You just have to prove that $\{s_n\}$ converges to an element of $U(I)$. The obvious candidate is $f+s_1$.

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