Physicists define path integrals by limits of discretizations, instead of using measure theory and Wiener measures.
Now, the issue is that most texts do not give clear definitions, so it seems almost everything is implicit in their discussions. Still, it seems worthwhile to understand what physicists are actually thinking when they do this, because most physicists texts are written in this language instead of the measure theoretic one.
I have then tried to guess what discretizing a functional means in this context, with a proper definition. So let $C^0([a,b],\mathbb{R}^d)$ be the space of continuous paths in $\mathbb{R}^d$ defined in $[a,b]$. I tried the following:
Definition (1): We say that a functional $\mathcal{F} : C^0([a,b],\mathbb{R}^d)\to \mathbb{C}$ depends on finitely many points if there is a partition $P = \{t_0,\dots, t_K\}$ of $[a,b]$ and a continuous $F : \mathbb{R}^{(K+1)d}\to \mathbb{C}$ such that $$\mathcal{F}[\gamma]=F(\gamma(t_0),\dots, \gamma(t_{K})).$$
Definition (2): A discretization of a continuous functional $\mathcal{F} : C^0([a,b],\mathbb{R}^d)\to \mathbb{C}$ is a sequence $(\mathcal{F}_N)$ of functionals depending on finitely many points, such that given $N < M$ the meshes of the partitions satisfy $|P_N| > |P_M|$ and such that $(\mathcal{F}_N)$ converges pointwise to $\mathcal{F}$. In other words, for each $\gamma \in C^0([a,b],\mathbb{R}^d)$ we have $\mathcal{F}_N[\gamma]\to \mathcal{F}[\gamma]$ as $N\to \infty$.
Definition (1) seems standard, since I've found something similiar in the path integral book by Lapidus. Definition (2) is my guess. The condition on the partitions meshes would be to ensure that $N\to \infty$ corresponds to the continuum limit.
My question: is this the correct definition of discretization that underlies the Physicist approach? If not what would be the correct definition?
This is close to how physicists think of path integrals, but they often work in Fourier transform space. Thus the discretization would limit frequencies not points. Also most path integral calculations use a nearly quadratic Lagrangian, for which the formulae are closed form, and then derive a perturbation series to add non-linear corrections. Finally the Feynman-Kac theorem demonstrates the relationship between random walks and the diffusion equation---the imaginary time equivalent of path integrals and the Schrodinger equation.