Intereting Posts

$\lim_{x \to 0} \dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} = 1$ for any $f,g \in C^1$ that are tangent to $\text{id}$ at $0$ with some simple condition
Cofinality of $2^{\aleph_\omega}$
Find $\sum_{i\in\mathbb{N}}(n-2i)^k\binom{n}{2i+1}$
Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$
Question regarding Nested Interval Theorem
When is the union of topologies a topology?
Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?
Picture of a 4D knot
$K$ consecutive heads with a biased coin?
Does this “extension property” for polynomial rings satisfy a universal property?
Maps in Opposite Categories
An integral that might be related to the modified Bessel function of second kind
Use two solutions to a high order linear homogeneous differential equation with constant coefficients to say something about the order of the DE
Is $x/x$ equal to $1$
Drawing colored balls

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?

- Evaluate $\int _{ 0 }^{ 1 }{ \left( { x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 2 } \right) \sqrt { 4{ x }^{ 3 }+5{ x }^{ 2 }+10 } \; dx } $
- Banach Tarski Paradox
- References about Iterating integration, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$
- Find the closed form for $\int_{0}^{\infty}\cos{x}\ln\left({1+e^{-x}\over 1-e^{-x}}\right)dx=\sum_{n=0}^{\infty}{1\over n^2+(n+1)^2}$
- Is the integral over a component of a doubly continuous function continuous?
- Standard machine in measure theory
- Probs. 10 (a), (b), and (c), Chap. 6, in Baby Rudin: Holder's Inequality for Integrals
- Evaluating the limit $\lim_{n\to\infty} \left$
- Proving Riemann Sums via Analysis
- Proof that a function with a countable set of discontinuities is Riemann integrable without the notion of measure

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.

- Relationship between the zeros of a vector field and the fixed points of its flow
- A Modern Alternative to Euclidean Geometry
- Some question about extension of bounded linear operator
- Attempt on fractional tetration
- Why is $\sqrt{8}/2$ equal to $\sqrt{2}$?
- Injective linear map between modules
- What are some equivalent statements of (strong) Goldbach Conjecture?
- Neumann series in an incomplete normed algebra
- Integrate $I=\int_0^1\frac{\arcsin{(x)}\arcsin{(x\sqrt\frac{1}{2})}}{\sqrt{2-x^2}}dx$
- A reference for existence/uniqueness theorem for an ODE with Carathéodory condition
- Convergence of an infinite product $\prod_{k=1}^{\infty }(1-\frac1{2^k})$?
- Group question: only one element $x$ with order $n>1$, then $x\in Z(G)$
- What groups can G/Z(G) be?
- Continuum between addition, multiplication and exponentiation?
- Let $f:\mathbb{R} \to \mathbb{R}$ be a function, so that $f(x)=\lfloor{2\lceil{\frac x2}\rceil}+\frac 12 \rfloor$ for every $x \in \mathbb{R}$.