Discounted infinite-horizon utility integrability ($b>0,L\le U\le R⟹{\int }_0^{\infty }{e}^{-bt}|U(c_t)|dt<\infty$)

Finite lower and upper utility bounds together with a positive discount rate bound the absolute time integral by $\max (|L|,|R|)/b$. This justifies expectation and integration in marginal-utility verification. Bounded utility alone on an undiscounted infinite horizon does not prevent both positive and negative parts from having infinite integral.