A generalisation of multi-dimensional integral, written $$ \int \mathcal{D}\phi \;\mathrm{exp} \left (\mathcal{F} [\phi] \right ) $$ where $\phi$ ranges over some restricted set of functions from a measure space $X$ to some space with reasonably nice algebraic structure. The simplest example is the case where $$ \phi \in L^2[X,\RR] $$ and $$ F[\phi] = -\pi \int_X \phi^2(x) d\mu(x) $$ in which case it can be argued that the result is $1$ . The argument is by analogy to the Gaussian integral $\int_{\RR^n} dx_1\cdots dx_n e^{-\pi\sum x_j^2} \equiv 1$ . Alas, one can absorb the $\pi$ into the measure on $X$ . Alternatively, following Pierre Cartier and others, one can use this analogy to define a measure on $L^2$ and proceed axiomatically.

One can bravely trudge onward and hope to come up with something, say à la Riemann integral, by partitioning $X$ , picking some representative of each partition, approximating the functional $F$ based on these and calculating a multi-dimensional integral as usual over the sample values of $\phi$ . This leads to some integral $$ \dotsint d\phi(x_1) \cdots d\phi(x_n) e^{f(\phi(x_1),\ldots,\phi(x_n))}. $$ One hopes that taking successively finer partitions of $X$ will give a sequence of integrals which converge on some nice limit. I believe Pierre Cartier has shown that this doesn't usually happen, except for the trivial kind of example given above.

The Feynman path integral was constructed as part of a re-formulation of quantum field theory by Richard Feynman, based on the sum-over-histories postulate of quantum mechanics, and can be thought of as an adaptation of Green's function methods for solving initial/boundary value problems. No appropriate measure has been found for this integral and attempts at pseudomeasures have given mixed results.

Remark: Note however that in solving quantum field theory problems one attacks the problem in the Feynman approach by `dividing' it via Feynman diagrams that are directly related to specific quantum interactions; adding the contributions from such Feynman diagrams leads to high precision approximations to the final physical solution which is finite and physically meaningful, or observable.

Bibliography

1

Hui-Hsiung Kuo, Introduction to Stochastic Integration. New York: Springer (2006): 250 - 253

2

J. B. Keller & D. W. McLaughlin, ``The Feynman Integral'' Amer. Math. Monthly 82 5 (1975): 451 - 465