Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting $\lambda = \mu - \nu$. Let $f$ be an extended real-valued $\mathfrak{A}$-measurable function on $X$.
Consider the following equality, which is known to hold whenever both sides of the equation exist (which happens, in particular, when $f$ is bounded). $$ \int_X f\ d\lambda = \int_X f\ d\mu - \int_X f\ d\nu \tag{*} $$ The integral on the left is by definition $\int_X f\ d\lambda^+ - \int_X f\ d\lambda^-$, where $\lambda = \lambda^+ - \lambda^-$ is the unique Jordan decomposition of $\lambda$.
It can be shown that it is possible for the left side of the equation to exist (and be finite) while the right side is undefined (i.e. the right side is of the form $\infty - \infty$).
What about the converse? Is it possible for the right side to exist in $\mathbb{R} \cup \{\pm\infty\}$ while the left side is undefined? Is it possible for the right side to exist in $\mathbb{R}$ while the left side is undefined?