calculus – Question about Integrability definition using Partitions of Unity

On page 65 of Spivak’s Calculus on Manifolds, when defining the integral of an (potentially unbounded) $$f$$ on an (potentially unbounded) set $$A$$, there is a theorem, which states:

If $$A$$ is Jordan-measurable and $$f$$ is bounded, then this definition
of $$int_A f$$ agrees with the old one.

Note that “this definition” refers to $$int_A f = sumlimits_{phi in Phi} phi cdot f$$, where $$Phi$$ is a partition of unity for some admissable open cover of $$A$$. The “old definition” refers to the integral defined on a bounded set and a bounded $$f$$ on this set, using $$U(f,P)$$ and $$L(f,P)$$.

Question:
$$A$$ is not stated to be bounded in this theorem. Is this just an omission, or does this follow from $$A$$ being Jordan-measurable?