# Tagged Questions

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### About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {${x∈X∣lim f_{n}∈R}$}  = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)} and that ...
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### Some properties of points with Lebesgue density equal to $1$.

I am studying Evans-Gariepy book and in corollary 1 of section 3.1.2, he prove that if $f:\mathbb{R}^N\to\mathbb{R}^M$ is locally Lipschitz and $$Z=\{x:\ f(x)=0\},$$ then $Df(x)=0$ a.e. $x\in Z$. He ...
Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ... 1answer 65 views ### Differentiation under the integral if and only if we have an L^1 dominator Let f(x)\in L^2(\mathbb{R}) and define$$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$for t\geq0. We want to show that g(t) is continuously differentiable if and only if xf(x)\in ... 1answer 32 views ### The functions \{f_n(x) = n\} are analytic and each miss the points -2, -3. But, they are not a normal family. So what am I missing. Thanks. Here is a theorem of Montel: Let \mathcal{F} be a family of analytic functions defined on a domain \Omega . If there are two fixed complex numbers a and b that are omitted from the range of ... 0answers 43 views ### Prove that f\ast g is defined a.e., integrable, and such that ∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1 Let f,g : \mathbb{R} → \mathbb{R} be L_1-functions. Set h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y). Prove that h(x) is defined a.e., h ∈ L_1(\mathbb{R}) and ∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1. So I ... 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 43 views ### Let (X,μ) be a measure space. Find a necessary and sufficient condition on (X,μ) that L_q(E) ⊂ L_p(E) for all 1 ≤ p < q ≤ ∞. Let (X,μ) be a measure space. Find a necessary and sufficient condition on (X,μ) that L_q(E) ⊂ L_p(E) for all 1 ≤ p < q ≤ ∞. I want to say that the condition is that E is finite. This ... 0answers 36 views ### ￼Show that \int_E f (x, y) dx is differentiable with respect to y and \frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx. Assume that f = f(x,y) is a function defined on E × (a,b). For each fixed y ∈ (a,b), f is integrable with respect to x on E, and for each fixed x ∈ E, f is differentiable with respect ... 0answers 14 views ### Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure? It is well-known that, L^{1}(\mathbb R) can be embed into M(\mathbb R) (= The space of complex Borel measure on \mathbb R); by identifying f\in L^{1}(\mathbb R) with the measure d\mu= f dm. ... 1answer 25 views ### Question on x-section of measurable rectangle in product measure space X \times Y I'm reviewing my analysis notes. We have that (X, \Sigma, \mu) and (Y, \tau, \nu) are complete measure spaces. We are considering the product measure space (X \times Y, \Sigma(\lambda^{*}), ... 2answers 55 views ### Borel measure supported on \mathbb{Q} Let \mu be a Borel measure supported on \mathbb{Q} \subset \mathbb{R}. Must \mu be a sum of Dirac measures? 2answers 36 views ### For 1 \leq r < p < \infty prove the continuous injection of L^p([0, 1]) into L^r([0, 1]). For 1 \leq r < p < \infty prove the continuous injection of L^p([0, 1]) into L^r([0, 1]). I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ... 1answer 42 views ### Let f_n:\mathbb{R}\rightarrow [0, 1] be functions such that \sup_{x \in \mathbb{R}}f_n(x) = 1/n and ||f||_1 = 1. Let f_n:\mathbb{R}\rightarrow [0, 1] be functions such that \sup_{x \in \mathbb{R}}f_n(x) = 1/n and ||f_n||_1 = 1. Set F(x) = \sup_{n \in \mathbb{N}}f_n(x). Prove that \int_\mathbb{R}F(x)dx ... 0answers 48 views ### Image of Cantor set under Cantor-Lebesgue function Let m^{\ast} be the Lebesgue outer measure and m the Lebesgue measure. Let \phi be the Cantor Lebesgue function and let \psi(x) := x + \phi(x). Let C be the standard Cantor set, why does ... 1answer 37 views ### Approximation functions in L^{1} by indicator functions of dyadic cubes Let \mu be a finite positive regular Borel measure on \mathbb{R}^{d} and let S be the family of finite unions of squares of the form \{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ... 1answer 45 views ### Unclear inequality in the proof of Birkhoff ergodic theorem. I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ... 1answer 39 views ### If A has positive Haar measure then AA^{-1} is a neighborhood of e I read the following exercise: Prove that if G is a locally compact topological group with Haar measure \mu and A \subset G, \mu (A) >0, then AA^{-1} contains an open neighborhood of the ... 1answer 63 views ### \int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C for some constant C > 4. Suppose f and g are nonnegative measurable functions on the interval [0,1], with the properties$$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$for some ... 0answers 52 views ### If f_{n}\rightharpoonup \bar{f} and f_{n}(x) \rightarrow f(x) pointwise a.e., then is \bar{f} = f a.e.? [duplicate] Suppose f_{n} is a sequence of functions in L^{p}(\mathbb{R}^{d}) such that \|f_{n}\|_{L^{p}} \leq 1 for all n and f_{n}(x) \rightarrow f(x) pointwise almost everywhere as n \rightarrow ... 2answers 27 views ### Show that \int_X gdν=\int_X gfdμ for all g∈L_1(ν). Let μ and ν be finite (positive) measures on a measurable space (X, M), and suppose that ν(E)=\int_E fdμ, for all E∈M, E where f is some function in L_1(μ). Show that \int_X ... 1answer 36 views ### Locally integrable function with a uniform bound… I'm a bit lost... I have a measure space (\Omega,\mathcal{B}(\Omega),\mu) where \mathcal{B}(\Omega) is a Borel set. Let f be a real-valued measurable function on \Omega and \mathcal{K} be ... 1answer 35 views ### Clarification about a basic proposition about measurable functions I am making my way through "Linear Functional Analysis" by Bryan P.Rynne and Martin A.Youngson (second edition). Given a measure space (X,\Sigma ,\mu ) we define a function f to be measurable if ... 2answers 49 views ### Can any measure be made into a bounded measure? Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure? 0answers 46 views ### extending the convergence of measures Let F the real vector space of all applications \phi: X \times Y \rightarrow \mathbb{R} where (X,\mathcal{B}_1, \mu), (Y,\mathcal{B}_2 ) measurable spaces with X and Y are compact metric ... 2answers 46 views ### Show that f is measurable Let U be a open Set of \mathbb{R} \times [0,\infty] and let f be defined as$$f: \mathbb{R}\mapsto [0,\infty], \quad f(x) := \max\{0,\sup\{y| (x,y) \in U\}\} $$How can I show that f is ... 0answers 135 views ### Why is the derivative of the translates of a measure measurable? Let G be a topological group and X a measure space. Let G \times X \rightarrow X be a measurable group action, \mu a \sigma-finite measure on X, and g\mu (for any g \in G) the measure ... 1answer 143 views ### E \subseteq [0, 1], m(E) > 0. Show that there are \alpha and \beta such that \alpha, \alpha + \beta, \alpha + 2\beta \in E. This was originally a proof verification question, but I have since moved the proof to an answer as discussed on meta. I still welcome comments on the proof as well as any alternative proofs. ... 0answers 77 views ### Is Fourier transform density preserving? I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ... 2answers 61 views ### Is it true that f \in L_1([a,b]) is the uniform limit of polynomials? Is it true that f \in L_1([a,b]) is the uniform limit of polynomials? And why? I know it is the uniform limit on a set take out some finite measurable set but not sure if I can say more. Thanks. 1answer 29 views ### Does there exists f\in \mathcal{S} (\mathbb R) so \hat{f}=1 on a comapct set C and \hat{f}=0 outside C\subset W (open set)? Let C is a compact subset of \mathbb R, V\subset \mathbb R, and 0<m(V)<\infty, where m is a Lebsgue measure on \mathbb R. My Question is: Can we expect to find k\in ... 1answer 67 views ### An amazing inequality of the integration of two functions. Let f:[0,1]\longrightarrow\mathbb{R} be measurable and g\in L^1[0,1] such that for all t>0,$$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$Prove that for 1<p<2,$$ ... 2answers 39 views ### For a real valued function$f(x,y)$on$\mathbb{R}^2$which is in$L_2$, show that$f(x+ε,y+ε) → f(x,y)$in$L_2$when$ε → 0.$[duplicate] For a real valued function$f(x,y)$on$\mathbb{R}^2$which is in$L_2$, show that$f(x+ε,y+ε) → f(x,y)$in$L_2$when$ε → 0.$Not sure how to go about this problem. I tried Fubini. But that ... 0answers 15 views ### For$k ∈ \mathbb{Z}$and for$x ∈ [k/n, (k + 1)/n)$set$g_n(x) = n\int_{k/n}^{\frac{k + 1}{n}}f(x)dx$. Let$f ∈ L_1(\mathbb{R}).$For$n ∈ \mathbb{N}$define the function$g_n :\mathbb{R}→\mathbb{R}$as follows. For$k ∈ \mathbb{Z}$and for$x ∈ [k/n, (k + 1)/n)$set$g_n(x) = n\int_{k/n}^{\frac{k + ...
Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...