This paper investigates generalized Hewitt-Stromberg (H-S) measures, focusing on how they behave in product sets. These sets are part of a special type of space called separable metric spaces. The goal is to understand inequalities that appear when working with such products. By introducing a pseudo H-S measure derived from pseudo-packing methods, we establish new product inequalities that do not require the doubling condition on component measures or dimension functions. These results extend classical inequalities in Euclidean spaces, providing a versatile framework for analyzing product measures in fractal geometry.