Consider the random variable $X: \Omega\rightarrow \mathbb{R}^l$ defined on the probability space $(\Omega, \mathcal{A}, P)$, with image $\mathcal{X}\subseteq \mathbb{R}^l$.
Consider the measurable space $(\mathbb{R}^l, \mathcal{B}(\mathbb{R}^l))$ where $ \mathcal{B}(\mathbb{R}^l)$ is the Borel sigma-algebra on $\mathbb{R}^l$.
Consider a function $f:\mathcal{X}\rightarrow \mathbb{R}$.
Suppose that we are said that $\int_{\mathcal{X}}f(t)d\mu$ is well defined, where $\mu$ is a measure on $(\mathbb{R}^l, \mathcal{B}(\mathbb{R}^l))$ [when $\mu$ is a probability measure then $\int_{\mathcal{X}}f(t)d\mu=\mathbb{E}(f(X))$].
I'm confused about the words well defined.
When $l=1$ and $\mathcal{X}=[a,b]$, the integral $\int_{[a,b]}f(t) dt$ exists (i.e. is well-defined) when
(i)$f$ is bounded on $[a,b]$
(ii) $\sup_PL_f(P)$ exists, where $L_f(P)$ is the Lower Riemann sum and P is a partition of $[a,b]$
(iii) $\inf_p U_f(P)$ exists, where $U_f(P)$ is the Upper Riemann sum and P is a partition of $[a,b]$
(iv) $\sup_PL_f(P)=\inf_p U_f(P)$
If I have understood correctly, just by writing down $\int_{[a,b]}f(t) dt$ it means that $\int_{[a,b]}f(t) dt$ is well-defined and, hence, $-\infty< \int_{[a,b]}f(t) dt<\infty$.
I guess we can generalise this definition to $l>1$ and $\mathcal{X}\subseteq \mathbb{R}^l$.
Question: Is there something similar for $\int_{\mathcal{X}}f(t) d\mu$? What does it mean that $\int_{\mathcal{X}}f(t) d\mu$ is well-defined (i.e. exists)? Does it imply that $-\infty<\int_{\mathcal{X}}f(t) d\mu<\infty$? And, is there any relation between the existence of $\int_{\mathcal{X}}f(t) dt$ and $\int_{\mathcal{X}}f(t) d\mu$?
My intuition is that, differently from case (1), writing down $\int_{\mathcal{X}}f(t) d\mu$ does not mean that $-\infty<\int_{\mathcal{X}}f(t) d\mu<\infty$, otherwise it wouldn't make sense assuming $\mathbb{E}(X)<\infty$ (when $\mu$ is a probability measure) as I have seen doing in many proofs. Could you clarify these points? I'm really confused. Any hint would be really appreciated.