Intereting Posts

How to compute the distance of an affine space from the origin
How does scaling $\Pr(B|A)$ with $\Pr(A)$ mean multiplying them together?
Another limit task, x over ln x, L'Hôpital won't do it?
Can we make $\tan(x)$ arbitrarily close to an integer when $x\in \mathbb{Z}$?
Connectedness of the boundary
Stirling numbers, binomial coefficients
Equality condition in Minkowski's inequality for $L^{\infty}$
Numbers as sum of distinct squares
When does the set enter set theory?
Is non-standard analysis worth learning?
For all graphs $\alpha(G) \ge \sum\limits_{x \ V(G)} \frac{1}{deg(x) + 1}$
Integral Inequality $\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$
Quadratic subfield of cyclotomic field
Laplacian Boundary Value Problem
How prove this $BC$ always passes through a fixed point with $\frac{x^2}{4}+y^2=1$

Let $f(t,x):(0,\infty) \times \mathbb{R}^d \rightarrow \mathbb{R}$ be a bounded and $\mathcal{B}([0,\infty))\otimes \mathcal{B}(\mathbb{R}^d)$-measurable function. Is the function

$$g(t,x) = \int_0^t f(s,x) ds $$

a $\mathcal{B}([0,\infty))\otimes \mathcal{B}(\mathbb{R}^d)$-measurable function? Why?

- Integral Inequality $\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$
- Integration of a trigonometric function
- Prove or disprove a claim related to $L^p$ space
- Sum of sets of measure zero
- Why do we essentially need complete measure space?
- Evaluating the integral $\int_0^\infty \frac{x \sin rx }{a^2+x^2} dx$ using only real analysis
- Area enclosed between the curves $y=x^2$ and $y=60-7x$
- Prove the following equality: $\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi$
- Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$
- What's the nth integral of $\frac1{x}$?

I will answer your first question. Your second question is not well-defined, as some of the commenters have noted. Returning to your first question, without any assumptions whatsoever on the measure spaces, I think the answer is no, but under some very weak assumptions – in particular that the two measure spaces in question are $\sigma$-finite, and that the integral of the function in question is finite – this is part of the content of the Fubini/Tonelli theorems for integration on product measures. So for your first question, since you’re considering $\mathbb{R}^n$ and Borel measures, then the answer is yes so long as the integral is finite.

To see this, we first prove a result about measurable sets. If $(X,\Sigma_1,\mu), (Y,\Sigma_2,\nu)$ are the measure spaces, with $A \in \Sigma_1$ and $B \in \Sigma_2$ of finite measure, then take $E = A \times B$. If $E(x)$ and $E(y)$ be the “slices” of $E$ given by fixing $x$ and $y$, respectively, then $\nu(E(x)) = 1_A(x)\nu(B)$ and $\mu(E(y)) = \mu(A) 1_B(y)$ are measurable in $X$ and $Y$, respectively. By $\sigma$-finiteness, then we can take increasing sequences of products $A_i \times B_i$ to show that the same result holds for $A,B$ measurable. So this proves the desired result for when $f$ is a characteristic function. Then, to obtain the general result for measurable $f(x,y)$ on $X \times Y$, we use the fact that $f$ can be approximated pointwise by a sequence of simple functions.

I haven’t actually proven (only sketched) my claims above. For more details, you can see any analysis textbook on integration.

- Polynomial Question
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- Indefinite integral of secant cubed
- Probability distribution for the perimeter and area of triangle with fixed circumscribed radius
- Expectation of the maximum of i.i.d. geometric random variables
- Why does $\arctan(x)= \frac{1}{2i}\log \left( \frac{x-i}{x+i}\right)+k$?
- 2 Tricks to prove Every group with an identity and x*x = identity is Abelian – Fraleigh p. 48 4.32
- Sum of $\lfloor k^{1/3} \rfloor$
- Partitions of 13 and 14 into either four or five smaller integers
- Stability for higher dimensional dynamical systems
- The converse to Schwarz Pick lemma?
- Parametric Form for a General Parabola
- Deriving the rest of trigonometric identities from the formulas for $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, and $\cos (A-B)$