Let $(\Omega, \mathcal F,\mu)$ be a measurable space. Let $f:\Omega\rightarrow \mathbb R$ be $\mathcal F$-measurable.

We know that: $\int_\Omega |f|d\mu<\infty\implies\int_\Omega fd\mu<\infty $.

If $\int_\Omega fd\mu<\infty$, then this means the function is integrable, and both $\int_\Omega f^+d\mu<\infty$ and $\int_\Omega f^-d\mu<\infty$, which implies that $|f|=f^++f^-$ is also integrable.

If both conditions are equivalent, then why do we use the absolute value. Why not spare notation?

Let $(\Omega, \mathcal F,\mu)$ be a measurable space. Let $f:\Omega\rightarrow \mathbb R$ be $\mathcal F$-measurable.

We know that: $\int_\Omega |f|d\mu<\infty\implies\int_\Omega fd\mu<\infty $.

If $\int_\Omega fd\mu<\infty$, then this means the function is integrable, and both $\int_\Omega f^+d\mu<\infty$ and $\int_\Omega f^-d\mu<\infty$, which implies that $|f|=f^++f^-$ is also integrable.

If the reverse holds, then both conditions are equivalent, then why do we use the absolute value. Why not spare notation?

Let $(\Omega, \mathcal F,\mu)$ be a measurable space. We know that: $\int_\Omega |f|d\mu<\infty\implies\int_\Omega fd\mu<\infty $.

Does the reverse hold? If not, can you provide an example?

Let $(\Omega, \mathcal F,\mu)$ be a measurable space. Let $f:\Omega\rightarrow \mathbb R$ be $\mathcal F$-measurable.

We know that: $\int_\Omega |f|d\mu<\infty\implies\int_\Omega fd\mu<\infty $.

If $\int_\Omega fd\mu<\infty$, then this means the function is integrable, and both $\int_\Omega f^+d\mu<\infty$ and $\int_\Omega f^-d\mu<\infty$, which implies that $|f|=f^++f^-$ is also integrable.

If both conditions are equivalent, then why do we use the absolute value. Why not spare notation?