Transform an integral from Hausdorff to Lebesgue

Question

Suppose I have the following integral with respect to the $(N-1)$-dimensional Hausdorff measure $$ \int_{\Gamma} g(x) \,\,d\mathcal{H}^{N-1}(x) $$ where $$ \Gamma = \left\{x\in \mathbb{R}^N\, :\, f(x) = c\right\} $$ How can I transform this integral into a Lebesgue integral? Ideally I would like to have something like this $$ \int_{\mathbb{R}^N} h(x) d\lambda^N(x) \qquad \text{or} \qquad \int_{\mathbb{R}^{N-1}} h(x) d \lambda^{N-1}(x) $$