# Tagged Questions

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### Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
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### Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$\int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty$$ if and only if $p=2$. That ...
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### Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
Let $\Omega$ be a measurable space with measurable sets $\Sigma$ and denote the space of simple functions by:$$\mathcal{S}:=\{s:\Omega\to\mathbb{R}:s=\sum_{k=1}^K ... 1answer 27 views ### Measurability of Modulus Context: This problem came up while reading an essay on Bochner integrability. Let \Omega be a measure space and E a Banach space. Consider two plain functions f:\Omega\to E and g:\Omega\to ... 0answers 19 views ### Integration by parts for multidimensional Lebesgue-Stieltjes Integrals I am concerned with the following problem: I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given ... 1answer 50 views ### Property of the Riemann Integral Hello fellow Mathematics enthusiasts. I was hoping someone could help me with the following problem from Terry Tao's Introduction to Measure Theory: Let [a,b] be an interval, and let f,g:[a,b] \to ... 2answers 49 views ### Extending Positive Functionals: Linearity How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set:$$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$and a linear subspace ... 2answers 47 views ### Bochner: Absolute Integrability For a Bochner measurable function it holds:$$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$for any positive measure \lambda\geq 0. The one direction is relatively simple when ... 1answer 89 views ### What is an example of a function that is measurable but not strongly measurable? Let (\Omega, \Sigma) be a measurable space and X a Banach space. Let f: \Omega \rightarrow X. f is called measurable if every the preimage of every Borel set in X is an element of ... 1answer 29 views ### About a \sigma-finite measure Consider a probability space (\Omega,\mathcal H,P) and a real random variable X such that E(X) is well defined (also infinite values are allowed). Is it true that the measure ... 1answer 32 views ### From \left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon show convergence a.e. of the series. I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the L^2-norm. In short, I just don't have any intuition for it. And I don't ... 1answer 32 views ### Continuity of measure and integration Suppose that f is a measurable function (\Omega, \mathfrak{F}, \mu) such that \int_{A}f \, d\mu \geq 0 \forall A \in \mathfrak{F}. Prove that f \geq 0 \ \mu-almost surely. Hint: Let A_n = ... 1answer 45 views ### Halmos Measure Theory section 39 Theorem D I have trouble explaining the remark "The function \phi plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals". I know ... 0answers 31 views ### Derivation using Ito calculus? I am reading the paper "Coupling Wiener processes by using copulas" by P. Jaworski and I've come across a statement I cannot reproduce. Let L^{-} and L^{+} be differential operators acting on ... 0answers 35 views ### Projection measures and integrability Let (M, \mathcal{A}, \mu) a probability space, Y compact metric space. Consider \mathcal{M}(\mu) be the space of probability measures \eta on M\times Y such that \pi_{*}\eta=\mu  where ... 1answer 18 views ### Reference on Riesz representation theorem for L^p(0,T,X) spaces. Brezis Functional Analysis book proves the following Riesz representation theorems for usual L^p(\Omega) spaces: In what book can we find an analogous of these theorems for L^p(0,T,X) spaces? ... 1answer 136 views ### Separability of a set with norm \thickapprox L^1 +L^{\infty} Let (M, \mathcal{A}, \mu) a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ... 2answers 56 views ### Is a L^1-function which is linear near the origin in L^p? Suppose you have a function f on \mathbb{R}, such that$$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$and$$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$for u \to 0. ... 1answer 45 views ### Series converges but term by term integration not valid? Give an example of a series \sum g_n of Lebesgue integrable functions on \mathbb{R} that converges but for which term by term integration is not valid. This is last minute exam revision so I do ... 1answer 23 views ### What is the integral of \int_{\mathbb{N}} s d\mu where \mu is the counting measure on \mathbb{P}(\mathbb{N})? What is the integral of \int_{\mathbb{N}} s d\mu where \mu is the counting measure on \mathbb{P}(\mathbb{N})? What does it mean for s to be integrable? 1. This is last minute exam revision. ... 1answer 57 views ### Do we need \mu, \nu to be \sigma-finite to show \int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu? The problem statement: Let (X, \mathcal F, \mu), (Y, \mathcal G, \nu) be \sigma-finite and f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu). Show that fg \in \mathcal L^1 (\mu \otimes ... 1answer 69 views ### How to understand uniform integrability? From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me? 1answer 42 views ### An issue with \infty \cdot 0 in showing that Cartesian product of a set with a null set has measure zero Here is the problem: Let (X, \mathcal A, \mu) and (Y, \mathcal B, \nu) be \sigma-finite measure spaces. Furthermore A\in \mathcal A and N\in \mathcal B such that \nu(N)=0. Let ... 1answer 52 views ### If two measures are equal, are the integrals with respect to these measures equal? If \mu and \nu are probability measures such that \mu=\nu, then is it true that for all measurable function f$$\int fd\mu=\int fd\nu \ \ \ ?$$It is true for integrable functions but if  ... 1answer 440 views ### Reinventing The Wheel - Part 2: The Lebesgue Integral Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ... 1answer 68 views ### Negative part of the integrand in an iterated integral Hi everyone: Suppose that (X,\mathfrak{M},\mu) and (Y,\mathfrak{N},\nu) are two measure spaces and f(x,y) is an extended real valued measurable function on X\times Y. Suppose we can not apply ... 1answer 44 views ### The space C_c^\infty(\mathbb{Q}_p^*) of smooth compactly supported functions on \mathbb{Q}_p^* Let p be prime. Let \mathbb{Q}_p^* be the multiplicative group of the field of p-adic numbers. We call a function f:\mathbb{Q}_p^*\rightarrow\mathbb{C} smooth if it is invariant under ... 0answers 18 views ### Non Borel Spaces: Gauge Integral Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ... 1answer 37 views ### A tricky integral with vanishing domain I would love to have the following result, however I got no clue if it is even true! Let B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\} for some sequences ... 0answers 17 views ### Finiteness of the lower integral implies finiteness a.e. of the function I want to prove that if a function f is \mu-measurable, f\geq 0  \mu-a.e., then the integral of f exists, that is its upper and lower integrals coincide. I've found the proof in Modern and ... 1answer 44 views ### If f is +\infty on a set of positive measure and the integral exists in [-\infty,+\infty], must the integral be +\infty? Suppose (X,\mathcal{M},\mu) is a measure space and f a measurable function from X to [-\infty,+\infty]. Suppose that$$\int_{X}f\ d\mu$$exists in [-\infty,+\infty], and that X contains a ... 0answers 64 views ### Why is the value assigned to a gauge integral well defined (unique)? Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ... 1answer 36 views ### Finite Measure Space: Integral Closure = Bochner Integral I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let \Omega be a finite measure space and E a Banach ... 2answers 69 views ### Without Lebesgue Everyone knows following problem. Let f be positive function on [0,1] and there exist I = \int_{0}^{1}f(x)dx. Prove that I>0. (recall that there are only two cases: I=0 or I>0. NOT ... 1answer 32 views ### variation of a function over countable intervals Let f be a function of bounded variation on [0,1]. Let \{[a_n,b_n]\}_{n=1}^\infty such that (a_n,b_n) are pairwise disjoint and \cup_{n=1}^\infty [a_n,b_n]=[0,1]. (for example, [1/2, 1], ... 1answer 33 views ### Proving and visualizing \mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x) Here is a trick from one of the proofs in probability:$$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$for a>0. So ... 0answers 120 views ### What types of integrals cannot be solved using improper Riemann-Stieltjes Integration? I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ... 0answers 172 views ### Open problems in Federer's Geometric Measure Theory I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ... 1answer 51 views ### Defining the integral on an arbitrary metric space I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions C[a,b]. I ... 2answers 47 views ### Jordan measure zero discontinuities a necessary condition for integrability The following theorem is well known: Theorem: A function f: [a,b] \to \mathbb R is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ... 2answers 93 views ### Do we need the f,g \geq 0 condition for \int f \ d\mu = \int g \ d\mu? My lecture notes state the following corollary: Let f,g \in \mathcal M_\bar{{\mathbb R}} (that is, numerical measurable functions), f=g \mu-almost everywhere and f,g \geq 0. Then \int f \ ... 1answer 40 views ### Locally integrable function with a uniform bound… I'm a bit lost... I have a measure space (\Omega,\mathcal{B}(\Omega),\mu) where \mathcal{B}(\Omega) is a Borel set. Let f be a real-valued measurable function on \Omega and \mathcal{K} be ... 1answer 50 views ### Haar measure on G \times G, where G is compact Let G be a compact group. Let \mu' and \mu be the Haar measure on G \times G and G, respectively, and further such that \mu'(G \times G) = 1 and \mu(G)=1. Does it follow that \mu' = \mu ... 0answers 34 views ### Basic facts related to Haar measure I have a compact group G and continuous functions f_1, f_2 from G to \mathbb{C} and g: \mathbb{R} \rightarrow \mathbb{C}. I have two questions related to Haar meausure. Is it true that$$ ...
Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...