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 $L(f,P)$.

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