TITLE: Existence Lebesgue integral and Lebesgue integrability of a function
QUESTION [0 upvotes]: 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?

REPLY [0 votes]: (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.
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