Path integral vs. measure on infinite dimensional space Coming from a mathematical background, I'm trying to get a handle on the path integral formulation of quantum mechanics. 
According to Feynman, if you want to figure out the probability amplitude for a particle moving from one point to another, you 1) figure out the contribution from every possible path it could take, then 2) "sum up" all the contributions.
Usually when you want to "sum up" an infinite number of things, you do so by putting a measure on the space of things, from which a notion of integration arises. However the function space of paths is not just infinite, it's extremely infinite. 
If the path-space has a notion of dimension, it would be infinite-dimensional (eg, viewed as a submanifold of $C([0,t] , {\mathbb R}^n))$. For any reasonable notion of distance, every ball will fail to be compact. It's hard to see how one could reasonably define a measure over a space like this - Lebesgue-like measures are certainly out.
The books I've seen basically forgo defining what anything is, and instead present a method to do calculations involving "zig-zag" paths and renormalization. Apparently this gives the right answer experimentally, but it seem extremely contrived (what if you approximate the paths a different way, how do you know you will get the same answer?). Is there a more rigorous way to define Feynman path integrals in terms of function spaces and measures?
 A: The answer is: forget about it. :-) Currently, there is no satisfying mathematical formalization of the path integral.
Coming from a mathematics background myself, I was equally dismayed at this state of affairs. But I have come to terms with it, mainly due to the following historical observation: for several centuries, infinitesimal quantities did not have a satisfying formalization, but that didn't stop mathematicians from understanding and using them. We now have the blessing of Weierstraß' epsilons and deltas, but it is also a curse, since the infinitesimal quantities disappeared as well (outside of non-standard analysis). I would say that the path integral is a similar situation.
However, if you accept the path integral as a "figure of speech", then there are ways to formalize it to some extend. Namely, you can interpret it as an "integration" of the propagator, much like the exponential function is the solution of the differential equation $\dot y = Ay$. The propagator is the probability amplitude
$$ U(r,t; r_0,t_0) $$
of finding a particle at place and time $r,t$ when it originally started at place and time $r_0,t_0$. It is the general solution to the Schrödinger equation
$$ i\hbar \frac{\partial}{\partial t}U(r,t; r_0,t_0) = \hat H(r,t) U(r,t; r_0,t_0), \quad U(r,t_0; r_0,t_0) = \delta(r-r_0) .$$
Now, pick a time $t_1$ that lies between $t$ and $t_0$. The interpretation of the propagator as probability amplitude makes it clear that you can also obtain it by integrating over all intermediate positions $r_1$.
$$ U(r,t; r_0,t_0) = \int dr_1 U(r,t; r_1,t_1)U(r_1,t_1; r_0,t_0) $$
If you repeat that procedure and divide the time interval $[t,t_0]$ into infinitely many parts, thus integrating over all possible intermediate positions, you will obtain the path integral.
More details on this construction can be found in Altland, Simons. Condensed Matter Field Theory.
A: For quantum mechanics, there's really nothing unrigorous about the path integral.  You have to define it in Euclidean signature, but that's just the way life is with oscillatory integrals.  It has nothing to do with the fact that the path integral is infinite-dimensional.  Try to insert a set of intermediate states in the propagator $\langle q_f | e^{-iHt/\hbar}|q_i\rangle$ and you'll get an integral that's not absolutely convergent.  This expression is just fine if you sandwhich it inside a well-defined computation -- e.g., don't use singular wave functions for your initial and final states -- but if you want the expression to stand on its own, you have to provide some additional convergence information.  Usually what people do is observe that the unitary group of time translations is the imaginary boundary of an analytic semigroup.  The real part of this semigroup, $e^{-H\tau}$, has a rigorous path integral formula; it's the volume of a cylinder set.  The volume of a cylinder set is computed as the limit of cutoff path integrals of the form
$$\frac{1}{Z} \int_{F_{cutoff}} e^{-\frac{1}{\hbar} S_{effective}(\phi)} d\phi,$$
where $d\phi$ is Lebesgue/Haar/whatever measures on the finite-dimensional space of cutoff fields and $S_{effective}$ is a cutoff/lattice approximation to the continuum action.  Given such a measure, under reasonable conditions, you can analytically continue the correlation functions from Euclidean signature back to Minkowski.
For the record:  mathematicians are not tearing their hair out about this stuff.  It's cool.  We got it.  We -- and by "we", I mean a relatively small number of experts, not necessarily including myself -- can even handle 4d Yang-Mills theory in finite volume.  (What's hard is proving facts about the behavior of correlation functions in the IR limit.) 
A: There are two more rigorous definitions of the path integral, the latter of which hasn't been mentioned yet:


*

*Gauge integrals: Muldowney's book "A Modern Theory of Random Variation", Wiley, 2012 has a chapter on complex-valued Brownian motion (chapter 7). It is the path to stochastic integration via a notion called random variability. Starting point is the Henstock integral ("gauge" integral), a generalisation of the Riemann-Stieltjes integral. This is not an approach via measures on function spaces, though.

*Cylindrical measures ("promeasures"). The book "Analysis, Manifolds and Physics, Part I: Basics" by Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick, North-Holland 1996, has a short introduction to this approach in chapter IV.D, and there you can find many references. The construction starts with the definition of a projective system of finite quotient spaces. The promeasure is a projective system of measures on this projective system of quotient spaces. The general Wiener integral towards Feynman diagrams is in the problems and exercises section at the very end of the book.

A: Path integral is indeed very problematic on its own. But there are ways to almost capturing it rigorously.
Wiener process
One way is to start with Abstract Wiener space that can be built out of the Hamiltonian and carries a canonical Wiener measure. This is the usual measure describing properties of the random walk. Now to arrive at path integral one has to accept the existence of "infinite-dimensional Wick rotation" and analytically continue Wiener measure to the complex plane (and every time this is done a probabilist dies somewhere).
This is the usual connection between statistical physics (which is a nice, well-defined real theory) at inverse temperature $\beta$ in (N+1,0) space-time dimensions and evolution of the quantum system in (N, 1) dimensions for time $t = -i \hbar \beta$ that is used all over the physics but almost never justified. Although in some cases it was actually possible to prove that Wightman QFT theory is indeed a Wick rotation of some Euclidean QFT (note that quantum mechanics is also a special case of QFT in (0, 1) space-time dimensions).

Intermezzo
This is a good place to point out that while path integral is problematic in QM, whole lot of different issues enter with more space dimensions. One has to deal with operator valued distributions and there is no good way to multiply them (which is what physicist absolutely need to do). There are various axiomatic approaches to get a handle on this and they in fact do look very nice. Except that it's very hard to actually find a theory that satisfies these axioms. In particular, none of our present day theories of Standard model have been rigorously defined.

Anyway, to make the Wick rotation a bit clearer, recall that Schrödinger equation is a kind of diffusion equation but for an introduction of complex numbers. And then just come back to the beginning and note that diffusion equation is macroscopic equation that captures the mean behavior of the random walk. (But this is not to say that path integral in any way depends on the Schrödingerian, non-relativistic physics)
Others
There were other approaches to define the path-integral rigorously. They propose some set of axioms that path-integral has to obey and continue from there. To my knowledge (but I'd like to be wrong), all of these approaches are too constraining (they don't describe most of physically interesting situations). But if you'd like I can dig up some references.
A: In 2-dimensional space-time, Feynman path integrals are perfectly well-defined, though understanding how this is done rigorously is somewhat heavy-going. But everything is spelled out in the book ''Quantum Physics: A Functional Integral Point of View'' by Glimm and Jaffe.
http://www.amazon.com/Quantum-Physics-Functional-Integral-Point/dp/0387964770
In 4 space-time dimensions how to make rigorous sense of the Feynman path integral is an unsolved problem. On the one side, there is no indication that some rigorous version of it could not exist, and one expects that the structural properties the integral has in 2D continue to hold. On the other side, constructing a 4D integral having these properties has been successful only in the free case, and the methods used for the constructions in lower dimensions seem too weak to work in 4 dimensions.
Edit: On the other hand, in quantum mechanics with finitely many degrees of freedom, Feynman path integrals are very well understood, and whole books about the subjects have been written in a mathematically rigorous style, e.g., the book ''The Feynman Integral and Feynman's Operational Calculus'' by Johnson and Lapidus. http://tocs.ulb.tu-darmstadt.de/110841727.pdf
A: One major difficulty with defining path-integrals (which is entirely mathematicians' fault)is that the mathematicians insist for no good reason (and many bad ones) that there are non-measurable subsets of R. This is a psychological artifact of early days of set theory, where ZFC ws not seen as a way of generating countable models of a set-theoretic universe, but as the way things REALLY are in Plato-land (whatever that means). Cohen fixed that in 1963, but mathematicians still haven't gotten used to the fix, but that is changing rapidly.
If you assume every subset of R is measurable, the notion of "randomly picking a number between 0 and 1" becomes free of contradiction. In the presence of the axiom of choice, the question "what is the probability that this number lands in a Vitali set?" is paradoxical, but in the real world, it is obviously meaningful. This tension is resolved in what is called a "Solovay model", where you have no more trouble with probability arguments meshing with set theory. For a non-mathematician, when you deal with sets which arise by predicative definition, not by doing uncountable axiom-of-choice shenanigans, probability is never contradictory. A solovay model still allows you to use countable axiom-of-choice, and countable dependent choice, which is enough for all usual analysis.
Anyway, inside a Solovay model, you can define a Euclidean bosonic path integral very easily: it is an algorithm for picking a random path, or a random field. This has to be done by step-wise refinement, because the random path or random field has values at continuum of points, so you need to say what it means for a path to "refine" another path. further, while paths end up continuous, so that the refinement process is meaningful in the space of continuous paths, fields refine discontinuously. if you have a field whose average value on a lattice is something, it swings more wildly at small distances in dimensions higher than 2, so that in the limit, it defines a random distribution.
If you are allowed to pick at random, part of the battle is won. You get free field path-integrals in any dimension with absolutely no work (pick the fourier components at random as Gaussian random numbers with a width which goes like the propagator). There is no issue with proving measurability, and the space of distributions you get out is just defined to be whatever space of distributions you get by doing the random picking. It's as simple as that. Really.
The remaining battle is just renormalization, at least for bosonic fields with CP invariant (real) actions, which have a stable vacuum, so that their Euclidean continuation has a probability interpretation. You need to define the stepwise approximations in such a way that their probability distribution function approaches a consistent limit at small refinements. This is slightly tricky, but it automatically defines the measure if you have a Solovay world. There is nobody working on field theory in Solovay models, but there are people who mock up what is more or less the same thing inside usual set theory by doing what is called "constructive measure theory". I don't think that one can navigate the complicated renormalization arguments unless one is allowed to construct measures using probability intuitions without fear, and without work. And set theorists know how to do this since 1970. 
