# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
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### Following conditions for convergence of measures equivalent

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Let $\mu_n$ be a sequence of finite measures on $([0, 1], \mathcal{B})$ and let $\mu$ be another finite measure on $([0, 1], \mathcal{B})$. ...
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### Does it follow that two finite positive measures are the same?

Suppose $\mu$ and $\nu$ are finite positive measures on the Borel $\sigma$-algebra on $[0, 1]$ such that $\int f\,d\mu = \int f\,d\nu$ whenever $f$ is real-valued and continuous on $[0, 1]$. Does it ...
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### Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Here is a proof of the dominated convergence theorem. Theorem. Suppose that $f_n$ are measurable real-valued functions and $f_n(x) \to f(x)$ for each $x$. Suppose there exists a nonnegative ...
Suppose $f_n$, $g_n$, $f$ and $g$ are integrable, $f_n \to f$ almost everywhere, $g_n \to g$ almost everywhere, $|f_n| \le g_n$ for each $n$, and $\int g_n \to \int g$. Does it follow that $\int f_n \... 1answer 27 views ###$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$implies$\{f_n\}$is uniformly integrable? Suppose$\mu$is a finite measure and for some$\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that$\{f_n\}$is uniformly integrable? 1answer 53 views ### Does there exist a subsequence$n_j$such that$\int_A f_{n_j}(x)\,dx$converges for each Borel subset$A$of$[0, 1]$? Let$\{f_n\}$be a sequence of measurable real-valued functions on$[0, 1]$that is uniformly bounded. Does there exist a subsequence$n_j$such that$\int_A f_{n_j}(x)\,dx$converges for each Borel ... 1answer 25 views ### Does it follow that$\{f_n\}$is uniformly integrable? Suppose$\mu$is a finite measure,$f_n \to f$almost everywhere, each$f_n$is integrable,$f$is integrable, and$\int |f_n - f| \to 0$. Does it follow that$\{f_n\}$is uniformly integrable? 1answer 71 views ### Do we necessarily have that$\int g\,d\mu_n \to \int_0^1 g\,dx$? Let$\mathcal{B}$be the Borel$\sigma$-algebra on$[0, 1]$. Suppose$\mu_n$are finite measures on$([0, 1], \mathcal{B})$such that$\int f\,d\mu_n \to \int_0^1 f\,dx$whenever$f$is a real-valued ... 2answers 49 views ###$\{f_n\}$is uniformly integrable if and only if$\sup_n \int |f_n|\,d\mu < \infty$and$\{f_n\}$is uniformly absolutely continuous? Let$(X, \mathcal{A}, \mu)$be a measure space. A family of measurable functions$\{f_n\}$is uniformly integrable if given$\epsilon$there exists$M$such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)... 1answer 27 views ### Countable collection of Borel subsets of [0, 1], exists subsequence where \int_A f_{n_j}(x)\,dx converges for each i? Let \{f_n\} be a sequence of measurable real-valued functions on [0, 1] that is uniformly bounded. How do I see that if \{A_i\} is a countable collection of Borel subsets of [0, 1], then there ... 3answers 36 views ### How do I see that if A is a Borel subset of [0, 1], then there exists a subsequence n_j such that \int_A f_{n_j}(x)\,dx converges? Let \{f_n\} be a sequence of measurable real-valued functions on [0, 1] that is uniformly bounded. How do I see that if A is a Borel subset of [0, 1], then there exists a subsequence n_j ... 1answer 21 views ### seq. of nonneg. Lebesgue measurable functions on \mathbb{R}, have \limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx? Let f_n be a sequence of nonnegative Lebesgue measurable functions on \mathbb{R}. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, ... 0answers 32 views ### What does “Borel space”, unqualified, refer to? For examples of use, Google "in Borel space", without the quotes. I'm thinking it means either ℝ equipped with its Borel σ-algebra, or to Borel spaces in general (that is, topological spaces with a σ-... 1answer 18 views ### If two intervals are not disjoint, I can write them as a union of disjoint pieces In this online lecture, the professor writes:$$E_1 \cup E_2 = (\sqcup_{i=1}^n I_j) \cup (\sqcup_{k=1}^n J_k) = \sqcup_{i=1}^n \sqcup_{k=1}^n(I_j \cap J_k)$$where I_j, J_k are intervals of \... 1answer 16 views ### Is \sigma(X,Y) = \sigma(X, X \cdot Y) for two Random Variables X and Y? Suppose we have two real random variables X,Y. Then clearly $$\sigma(X, X \cdot Y) \subset \sigma(X,Y)$$ since both X and X \cdot Y are \sigma(X,Y)-measurable ... 1answer 69 views +50 ### Intuition behind proof of bounded convergence theorem in Stein-Shakarchi Theorem 1.4 (Bounded convergence theorem) Suppose that \{f_n\} is a sequence of measurable functions that are all bounded by M, are supported on a set E of finite measure, and f_n(x) \to f(x) ... 1answer 42 views ### f_n \to f almost everywhere and \int |f_n| \to \int |f| implies \int |f_n - f| \to 0? Suppose f_n and f are integrable, f_n \to f almost everywhere, and \int |f_n| \to \int |f|. Does it necessarily follow that$$\int |f_n - f| \to 0?$$1answer 25 views ### Integral of sequence converges? [closed] Suppose (X, \mathcal A, \mu) is a measure space, f and each f_n is integrable and nonnegative, f_n \to f almost everywhere, and \int_X f_n \to \int _X f. Does it necessarily follow that for ... 1answer 24 views ### Sequence of nonnegative f_n tending to 0 pointwise where \int f_n \to 0, but there's no integrable function where f_n \le g for all n? What is an example of a sequence of nonnegative functions f_n tending to 0 pointwise such that \int f_n \to 0, but there is no integrable function such that f_n \le g for all n? 1answer 15 views ### f integrable, if either A_n \uparrow A or A_n \downarrow A, then does it follow that \int_{A_n} f \to \int_A f? [closed] Suppose f is integrable. If either A_n \uparrow A or A_n \downarrow A, then does it follow that \int_{A_n} f \to \int_A f? 1answer 50 views ### Infinite dimensional Borel-measurable function. I am not quite sure how this statement about infinite dimensional borel-measurable functions is true, but the author says it is an easy observation: Let D([0,\infty)) denote the space of all ... 0answers 30 views ### What is an example of lower semicontinuous functions not satisfying this? Let X be a locally compact Hausdorff space and \mu be a Radon measure on X. Let u:X\rightarrow [0,\infty] be a lower semicontinuous function such that \int_X u d\mu <\infty. Then, does ... 0answers 7 views ### Can we approximate integrable functions by finite-valued upper,lower semicontinuous functions? Let X be a locally compact Hausdorff space and \mu be a Radon measure on X and f\in L^1(\mu) be real-valued and \epsilon>0. Then, by Vitali-Caratheodory theorem, there exist upper ... 0answers 27 views ### Exchanging supremum and conditional expectation I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process Y_t generating the filtration (F_t) and ... 3answers 68 views ### Show that there is a subsequence of (f_n)_n that converges to f almost everywhere. Let (X,\mathcal{B}, \mu) be a measure space and assume the sequence (f_n)_n converges to f in L^p(\mu), where 1\leq p<\infty. Show that there is a subsequence of (f_n)_n that converges ... 2answers 120 views ### sigma algebra generated by compacts versus sigma algebra generated by open sets Let \Omega be a locally compact Hausdorff set. Is the sigma algebra generated by compact sets is the same as the sigma algebra generated by open sets? 2answers 47 views ### L^1 convergence and limsup of convergent sequence I have to solve this exercise: let f_n be a sequence of positive real function defined on a measure space (X,M,\mu) such that f_n\in L^1(\mu) \forall n\in \mathbb{N} and f_n is convergent in ... 2answers 23 views ### If f,g:X \to Y are measurable, is the set on which f=g measurable? What Y does this hold for? If f,g:(X,\Sigma_X) \to (Y,\Sigma_Y) are measurable, when can we conclude that \{x \in X: f(x)=g(x)\} \in \Sigma_X is a measurable subset of X? This is a standard theorem when Y=\mathbb{R} or ... 0answers 60 views ### Question on Young's inequality for convolution Young's Inequality: Let u\in {\cal{L}^1}(\lambda^n) and v\in {\cal{L}^p}(\lambda^n) where \lambda^n is the n dimensional Lebesgue measure on (\mathbb{R}^n,{\cal{B}}(\mathbb{R}^n)) and p\in[... 1answer 58 views ### Real Analysis, Problem 3.2.14 The Radon Nikodym Theorem Problem 3.3.14 - If \nu is an arbitrary signed measure and \mu is a \sigma-finite measure on (X,M) such that \nu\ll \mu, there exists an extended \mu-integrable function f:X\rightarrow [-\... 1answer 37 views ### measure preserving map does not increase distance I read a sentence saying "any measurable subset of \mathbb{R} can be mapped to an interval by a measure-preserving transformation which does not increase distances" Here the measure is Lebesgue ... 2answers 56 views ### What is the infimum/supremum of a set in the extended real line? May be a simple question, but I still don't know what the definition of the supremum or infimum of a set in the extended real line is. For example when we define the Lebesgue measure, we define the ... 1answer 34 views ### Continuity and convergence everywhere. Suppose a sequence of continuous functions (h_n) converges almost everywhere to another continuous function h . Is it possible to infer that h_n infact converges everywhere? If not, under what ... 0answers 16 views ### Question with Lebesgue-Stieltges outer measure Let g and h be two increasing functions and \theta_g, \theta_h be the associated Lebesgue-Stieltges outer measures on R (the set of real numbers). We can also associate to g+h the L-S ... 1answer 44 views ### Real Analysis, Folland Proposition 3.11 The Radon Nikodym Theorem Proposition 3.11: If \mu_1,\ldots,\mu_n are measures on (X,M), there is a measure \mu such that \mu_j\ll\mu for all j -- namely \mu = \sum_1^n\mu_j. Attempted proof: Suppose we have a ... 2answers 42 views ### Real Analysis, Folland Corollary 3.10 The Lebesgue Radon Nikodym Theorem Background Information: Proposition 3.9 - Suppose that \nu is a \sigma-finite measure and \lambda are \sigma-finite measures on (X,M) such that \nu\ll \mu and \mu\ll \lambda. a.)... 2answers 13 views ### Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets Okay so I have a measurable function f and a set E_{n,i}$$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$where i=1,...,n2^{n} and n=1,2,... Then I have another ... 0answers 32 views ### Hölder inequality application to show that f=1 [closed] I want to proof that if f \in L^{1}_{\mu}(\mathbb{R}), f > 0 continuous, satisfies (\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{... 1answer 102 views ### Conclusion about measurable functions from knowledge about continuous functions Let \mu and \nu be two finite Borel measures on \mathbb{R}. We know that if$$\int f d\mu = \int f d\nu $$for all continuous functions f then \mu=\nu and so the equation above holds for ... 0answers 14 views ### Measure induced by subgradient of convex functional I am trying to understand why the following defines a measure. Let \phi : \mathbb{R}^d \to \mathbb{R} be a convex function. Define a measure \mu on \mathcal{B}(\mathbb{R}^d) by$$\mu(E) = \... 1answer 65 views ### Lebesgue integration by substitution I read that, if$f\in L^1[c,d]$is a Lebesgue summable function on$[a,b]$and$g:[a,b]\to[c,d]$is invertible and such that$g\in C^1[a,b]$and$g^{-1}\in C^1[a,b]$, then $$\int_\limits{g([a,b])}f(x)\... 1answer 20 views ### the relation between the sigma-algebras of two isomorphic spaces [closed] It crosses my mind the following question : if X and Y are two isomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between \... 0answers 15 views ### p-power integral and p-series in higher dimensions This seems like a basic question that should be addressed in a multivariable calculus course however I don't think I've ever confronted the issue until I became confused about a recent question here ... 4answers 888 views ### Is diameter of a set a measure? Suppose the diameter of a nonempty set A is defined as$$\sigma(A) := \sup_{x,y \in A} d(x,y)$$where$d(x,y)$is a metric. Is$\sigma(.)$a 'measurement'? I.e., how do I prove the countable ... 1answer 28 views ### Measure and probability : what would be$\mu(A\mid B)$? I know that a probability space$(\Omega ,\mathcal F,\mathbb P)$is in fact a measure space with a$\sigma -$algebra$\mathcal F$and a measure$\mathbb P$. I know that if$A,B\in \mathcal F$with$\...
Background Information: This is a Corollary to Theorem 3.5, found here. If $\mu$ is a measure and $f$ is an extended $\mu$-integrable function, the signed measure $\nu$ defined by \$\nu(E) = \int_{E}...