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.

 

[jvicens]

Hi there,

Integrals over $\mathbb R^{\mathbb R}$ and $\mathbb R ^{\mathbb N}$ can be defined using the theory of Lebesgue integration. This theory extends the concept of integration from finite-dimensional spaces to infinite-dimensional spaces, including function spaces such as $\mathbb R^{\mathbb R}$ and $\mathbb R ^{\mathbb N}$.

In order to define integrals over these spaces, we first need to define a measure on them. This measure assigns a numerical value to subsets of the space, and is used to determine the size or "volume" of these subsets. In the case of $\mathbb R^{\mathbb R}$ and $\mathbb R ^{\mathbb N}$, we typically use the Lebesgue measure.

Once we have a measure, we can then define the integral of a function over the space as the limit of a sequence of integrals over subsets of the space. This is similar to how integrals are defined in finite-dimensional spaces, but the main difference is that the subsets in the sequence are allowed to become arbitrarily small.

Regarding your question about the objects being integrated, in the theory of Lebesgue integration, the objects being integrated are functions rather than operators. However, there are other theories that extend the concept of integration to operators, such as the theory of operator-valued measures. These theories are also used in functional analysis.

I hope this helps answer your question. If you're interested in learning more, I would recommend looking into books on Lebesgue integration or functional analysis. Thank you for your curiosity and good luck with your studies!