I have a question related to the existence of Lebesgue Integral. Here in the paragraph "signed function", we read that the Lebesgue integral exists provided that $$(1) \min(\int_{E}f^+d\mu, \int_Rf^{-}d\mu)<\infty$$ Then we read that $f$ is Lebesgue integrable provided that $$(2) \int_R|f|d\mu<\infty$$
Which relation is between the two conditions?
My understanding is that: (2)--> (1)<-->Existence Lebesgue Integral, i.e. the Lebesgue integral exists even if a function is not Lebesgue integrable. Moreover (2) implies that the Lebesgue integral is finite. Is it correct?
Any hint would be really appreciated; I think my intuition is wrong because I found in many proof that a function $f$ can be Lebesgue integrable with a Lebesgue integral not necessarily finite. What is wrong in my argument?
(2) implies (1). When (1) holds and (2) does not, the Lebesgue integral exists but it is equal to $+\infty$ or $-\infty$. We distinguish this from the case when (2) holds through the two definitions.