Given an unsigned function $f : \mathbb{R}^d \to [0,+\infty]$, we can define lower and upper Lebesgue integrals by \begin{align*} \underline{\int_{\mathbb{R}^d}}f(x)\,dx &:= \sup_{0\leq g\leq f,\,g \text{ simple}} \operatorname{Simp}\int_{\mathbb{R}^d} g(x)\,dx \\ \overline{\int_{\mathbb{R}^d}}f(x)\,dx &:= \inf_{h \geq f,\,h \text{ simple}} \operatorname{Simp}\int_{\mathbb{R}^d} h(x)\,dx, \end{align*} assuming we've already defined the Lebegue integral for simple functions.
A result in Tao's Introduction to Measure Theory states:
If $f+g$ is a simple function that is bounded with finite measure support (i.e. is absolutely integrable) then $$ \operatorname{Simp}\int_{\mathbb{R}^d}(f(x)+g(x))\,dx = \underline{\int_{\mathbb{R}^d}} f(x)\,dx + \overline{\int_{\mathbb{R}^d}}g(x)\,dx. $$
I understand why we need to assume that $f+g$ is bounded to prove this (we need to be able to subtract simple functions from $f+g$), but why do we have to assume it has finite measure support?
Here's an attempt at a proof without the finite-measure support hypothesis, supposing we've already proven the result with that hypothesis:
Proof. If $\operatorname{Simp}\int_{\mathbb{R}^d}(f(x)+g(x))\,dx$ is finite, then $f+g$ has finite measure support and the result follows; thus it suffices to show that if $\operatorname{Simp}\int_{\mathbb{R}^d}(f(x)+g(x))\,dx$ is infinite, then at least one of $\underline{\int_{\mathbb{R}^d}} f(x)\,dx$ and $\overline{\int_{\mathbb{R}^d}}g(x)\,dx$ must be infinite.
We prove the contrapositive. Suppose that both quantities are finite. Then there exists a simple function $g \leq \psi \leq f+g$ such that $\operatorname{Simp}\int_{\mathbb{R}^d}\psi(x)\,dx$ is finite. Since $f+g$ is bounded, the function $\phi := f + g - \psi$ is defined, simple, and satisfies $0 \leq \phi \leq f$. Thus $\operatorname{Simp}\int_{\mathbb{R}^d}\phi(x)\,dx$ is finite, and since $f + g = \phi + \psi$, linearity of the simple integral implies that $\operatorname{Simp}\int_{\mathbb{R}^d}(f(x)+g(x))\,dx$ is finite.
Is this proof correct?
EDIT: Modified my question and provided more of my thought process.