Defining Integrals over R^R, R^N

[WWGD]

Hi,
Just curious: how does one define integrals over $\mathbb R^{\mathbb R}, \mathbb R ^{\mathbb N}$? I assume this must be a topic in Functional Analysis. I know a bit about abstract Wiener spaces; is there something else? And I assume the objects that are integrated are operators (linear, orat least continuous) between infinite dimensional spaces ( at least the domain of definition is infinite-dimensional). Hoping some one has a reference. Thanks.