In this note we deal with some problems related to the $H_1$-integral introduced recently by Garces, Lee, and Zhao. We give a~new definition of that integral and characterize $H_1$ integrable functions almost everywhere equal to zero. We also discuss some results stated in the original paper on the $H_1$-integral.
"On H₁-Integrable Functions." Real Anal. Exchange 27 (1) 275 - 286, 2001/2002.