Can the one-sided Laplacian transform be generalized to the distribution?

Question

My problem is that the one-sided Laplace transform I've encountered goes something like this. $$\int_0^{+\infty}f(t)e^{-st}dt$$ But I think it is against the strict definition of the generalized function that the lower bound of the integral of the unilateral Laplace transformation is 0 instead of negative infinity? Would it be more reasonable to define the unilateral laplace transform by Fourier transform as follows? $$\mathscr{L}T={\mathscr F}\{H(x)T(x)e^{-\sigma x}\}$$ $H(x)$ is Heaviside step funtion. I mainly want to know whether this formula is the definition of unilateral Laplace transformation under generalized functions