Let $(X,\mathcal{X},\mu)$ be a measure space.
Most definitions of the Lebesgue integral only work for
- simple functions $f = \sum_{n \in \mathbb{N}} \alpha_n 1_{S_n}$ (for measurable sets $S_n$ and $\alpha_n \in \mathbb{R}$) that satisfy $\sum_{n \in \mathbb{N}} \alpha_n \mu(S_n) < \infty$ and for
- measurable functions $f:X \to \mathbb{R}_{\geq 0}$ with $f=\lim_{n \to \infty} f_n$ for simple functions $f_n$ (with $\forall n \in \mathbb{N}:f_n \leq f_{n+1}$) with $\lim_{n \to \infty} \int f_n d\mu < \infty$
(As an example, see https://www.encyclopediaofmath.org/index.php/Lebesgue_integral)
My question: What happens if we expand the definition, allowing infinity in more places? In particular, I would like to allow
- simple functions $f = \sum_{n \in \mathbb{N}} \alpha_n 1_{S_n}$ (for measurable sets $S_n$ and $\alpha_n \in \mathbb{R}$) that satisfy $\sum_{n \in \mathbb{N}} \alpha_n \mu(S_n) \leq \infty$ and for
- measurable functions $f:X \to \mathbb{R}_{\geq 0} \cup \{ \infty \}$ with $f=\lim_{n \to \infty} f_n$ for simple functions $f_n$ (with $\forall n \in \mathbb{N}:f_n \leq f_{n+1}$) with $\lim_{n \to \infty} \int f_n d\mu \leq \infty$
Are there caveats when extending the definition in this way? A few properties that reasonably should still hold are
- $\int \sum_{n \in \mathbb{N}} f_n d\mu = \sum_{n\in \mathbb{N}} \int f_n d\mu$
- $\int \lim_{n\to \infty} f_n d\mu = \lim_{n \to \infty} \int f_n d\mu$ if $\forall n \in \mathbb{N}: f_n \leq f_{n+1}$
- Fubini: $\int_x d\mu \int_y d\mu' f(x,y) = \int_y d\mu' \int_x d\mu f(x,y)$ (for $\sigma$-finite measures $\mu,\mu'$ of course)
My thoughts: It seems to me that the crucial property to check is that for any measurable function $f$, no matter what $f_n$s we choose in $f = \lim_{n \to \infty} f_n$, we get the same result for $\lim_{n \to \infty} \int f_n d\mu$. Once we have that, the rest is simple.
Even Fubini becomes simple to prove, because we can write the measurable function $f(x,y)$ as $\sum_{n \in \mathbb{N}} f_n$ for simple functions $f_n$ with finite integral (see https://proofwiki.org/wiki/Measurable_Function_Pointwise_Limit_of_Simple_Functions), use linearity and apply Fubini in the common setting with finite integrals.
Note that my version of Fubini is stronger than what people usually use, because it does not require $\int_x d\mu \int_y d\mu' f(x,y) < \infty$ (I dropped the absolute values since my functions are positive anyways).