# Defining Integrals over R^R, R^N

• WWGD
In summary, integrals over ##\mathbb R^{\mathbb R}## and ##\mathbb R ^{\mathbb N}## can be defined using the theory of Lebesgue integration, which extends the concept of integration from finite-dimensional spaces to infinite-dimensional spaces. This involves defining a measure on the space and then taking the limit of a sequence of integrals over subsets of the space. These integrals are typically over functions, but there are also theories that extend integration to operators.
WWGD
Gold Member
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.

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!

## 1. What is an integral over R^R?

An integral over R^R is a mathematical concept used to find the area under a curve in a two-dimensional Cartesian plane, where the curve is defined by a function with real numbers as both the independent and dependent variables. It is also known as a double integral or a multiple integral.

## 2. How is an integral over R^R different from a regular integral?

An integral over R^R involves integrating a function of two variables over a region in a two-dimensional space, while a regular integral involves integrating a function of one variable over a one-dimensional interval. Additionally, an integral over R^R requires multiple integration techniques, such as iterated integrals, to solve.

## 3. What is the purpose of defining integrals over R^R?

The purpose of defining integrals over R^R is to calculate the total area under a curve in a two-dimensional space. This concept is used in various fields of science and engineering, such as physics, economics, and geometry, to find important quantities such as volume, mass, and probability.

## 4. What are some applications of integrals over R^R?

Integrals over R^R have numerous applications in science and engineering. They are used to calculate the volume of irregularly shaped objects, the center of mass of an object, the work done by a variable force, the probability of an event occurring in a given region, and many more.

## 5. Is there a specific method for solving integrals over R^R?

There is no specific method for solving integrals over R^R, as the approach may vary depending on the function and region of integration. However, common techniques include iterated integrals, change of variables, and using geometric interpretations. It is important to have a strong understanding of single variable integrals and basic multivariable calculus concepts when solving integrals over R^R.

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