Does an integral of the form $\int f(x) \, \sqrt{dx}$ have any meaning? If $f(x)$ is a Riemann-integrable function, what meaning is there to an integral of the form $$\displaystyle\int f(x) \, \sqrt{dx}~?$$ I have read that stochastic processes like Brownian motion may be described by integrals of somewhat unusual forms as $\displaystyle\int dW^2$. 
I suppose that $\displaystyle\int f(x) \, \sqrt{dx}$ doesn't represent a random process since $f(x)$ here is defined and deterministic. 
Any insight would be helpful.

It seems to me that you can look at this as the square-root of a differential $~1-$form ${(dx)}^{1/2}$ which has been asked and answered here:
What are half forms?
