# Integrating generalized functions / distributions with a particular property

We have a function $$f(x(y),y)$$ where $$x\in \mathbb{R}$$ depends on $$y\in \mathbb{R}^3$$, $$x$$ is a scalar but $$y$$ is a vector of $$\mathbb{R}^3$$ : \begin{align*} f(x,y)&= \frac{\partial}{\partial x} g(x,y) \\ \end{align*} For example $$x= x(y)= y\cdot v+ b$$

We want to consider cases where $$g$$ is discontinuous, so $$f$$ can be regarded as a distribution in the parameter $$x$$, that is,

$$\text{for } \phi\in S(\mathbb{R}): f(\phi)=\int_{\mathbb{R}} f(x,y)\phi(x)\,dx \text{ , for a fixed vector y}$$

We want to sum over the sphere, we know in the case when $$x$$ does not depend on $$y$$ we can do this :

We have the family of distributions $$\Big(f(x, y)\Big)_{y}$$ , to which we apply a notion of distribution-valued integral; the 'Gelfand-Pettis' weak integration, or Bochner integrals.
( under some conditions that $$f$$ is 'nice' enough as a function of $$y$$ ) \begin{align*} \left(\int_{S^2} f\,d\sigma(y) \right) (\phi)=& \int_{S^2} f(\phi)\,d\sigma(y) \quad \quad (\text{ Gelfand-Pettis }) \\ =& \int_{S^2} \left(\int_\mathbb{R} f(x,y)\phi(x)\,dx\right) \,d\sigma(y) \\ =& \int_{S^2}\int_\mathbb{R} - g(x,y)\phi'(x)\,dx \,d\sigma(y) \end{align*}

$$\textbf{My question is : Can we still do this when } x \textbf{ is a function of }y$$ ?

Is there another way to integrate these distributions ? in the sense that we want the integral to be a sort of generalized function of the smooth classical integral for when $$g$$ is continuous in $$x$$ .