# measurable

1. ### Borel Measurable Function

I need to show that f_{n} is Borel measurable for each n \in \mathbb{N} The solution shows for \alpha \textless 0 , \{x : f_{n}$$x$$ \textgreater \alpha \} = \[0, \infty \) 0 \leq \alpha \textless n , \{x : f_{n} $$x$$ \textgreater \alpha \} = \( \frac{ \alpha}{n^2}, \frac{2n -...
2. ### measurable and integrable of product space

Let f : (0,1) —>R be measurable( w.r.t. Lebesgue measure) function in L1((0,1)). Define the function g on (0,1)× (0,1) by g(x,y)=f(x)/x if 0<y<x<1 g(x,y)=0 if 0<x≤y<1 Prove: 1) g is measurable function (w.r.t. Lebesgue measure in the prodcut (0,1)× (0,1) 2)g is integrable in (0,1)× (0,1)
3. ### How can the measure of a measurable set be bounded?

Helle Everyone, In one book, the Lebesgue measure is said to possess the following properties (among others): 1) the measure of any measurable set can be approximated from above by open sets; that is, for any measurable M \mu (M) = inf \{ \mu(O): M \subset O, O is open \} 2) the measure...
4. ### [Real analysis] Is this function measurable?

[Real analysis] For x in |Rn, is the function f(x)=e-|x|^2 measurable? Why?
5. ### Jordan measurable set

http://www.tau.ac.il/~tsirel/Courses/Analysis3/lect6.pdf Hi, i need help with the solution of problem 6h3 (c),on page 87. in the attached link. for p not smaller than 1,i think i know,but there must be a nice and compact solution. Thanks in advance.
6. ### Lebesgue measurable

Prove that if f:R->([0,∞]) and f^(-1)((r,∞]) ∈ M for each r∈ M,then f is Lebesgue measurable. (M is the σ-algebra of Lebesgue measurable sets)
7. ### show that set is measurable

Hi all, I'm having trouble with this problem. Hint given is to show that A = { E: b-a = m*((a,b)∩E) + m*((a,b)~E) } is a sigma algebra. I understand the hint logic - if A is a sigma algebra then the sets in A are measurable sets so E must be measurable. I know that all open, bounded intervals...
8. ### sandwich measurable set

Suppose that A is subset of R (real line) with the property for every ε > 0 there are measurable sets B and C s.t. B⊂A⊂C and m(C\B)<ε Prove A is measurable By definition A is measurable we need to prove m(E)=m(E∩A)+m(E\A) for all E the ≤ is trivial enough to show ≥: Since C is measurable then...
9. ### outer measure of disjoint measurable sets question

I'm trying to prove that given a finite sequence of disjoint measurable sets, the outer measure of the union of the sets is equal to the sum of the outer measure of each set. This seems very obvious, I was thinking about using induction, but I keep getting stuck. Any thoughts would be much...
10. ### Measurable Functions

Hi, math geniuses, Im having trouble with these two real analysis problems. 1) F Measurable, h in R, prove that g(x) = f(x+h) is measurable. 2) f continuous in almost every point. => f is measurable. Thanks!
11. ### Lebesgue measurable functions

Just wondering about a proof here given in my lecturer's notes: show that cf : \mathbb{R} \to \mathbb{R} is Lebesgue measurable, where c is a scalar and f a function. I understand it in the cases where c=0 and c \textgreater 0, but in the case ofc \textless 0, this is the proof...
12. ### Function composition: measurable with non-measurable

True or false? If X is a random variable and f(X) is measurable (f not necessarily measurable), then there's a measurable function g such that f(X) = g(X)
13. ### Showing a function is measurable

Let X=(S,d) a banach space. I want to show the following: For any borel measure \mu, the map \psi: X\to \mathbb{R} given by x\mapsto \mu(\overline{B}_x(r)) is measurable. I believe this can be done in a number of ways, one of which is proving that the set U_c:=\left\{x:\psi(x)\leq c\right\}...
14. ### prove countable set is measurable?? plz,

plz. help me in that ,, (Worried) prove countable setis measurable ??? (Headbang) plz ASAP...(Crying)
15. ### measurable functions

Let f be a measurable function and g be a 1-1 function from R to R which has a Lipschitz inverse. Show that the composition fog is measurable My idea was to take a set (a,00) , then f^-1(a,00) is measurable call this set C , but i need to prove that g^-1(C) is measurable Since g^-1 is...
16. ### E-E for a Lebesgue measurable set E contains some box?

I want to show that if E\subset \mathbb{R}^d is a Lebesgue measurable set where\lambda(E)>0, then E-E=\{x-y:x,y\inE\}\supseteq\{z\in\mathbb{R}^d:|z|<\delta\} for some \delta>0, where |z|=\sqrt{\sum_{i=1}^d z_i^2}. MY approach is this. I take J\in\mathcal{E}, a box in \mathbb{R}^n with equal...
17. ### Measurable surds.

The data are positive rational numbers A and B, for which the number √a + √b + √a√b is measurable. Prove that the number √a and √b are also measurable. please make a quick and detailed task Really thanks for all the answers
18. ### Countable mutually disjoint measurable sets

I have an end of section problem from my analysis book that I would like a hint on, it reads: Prove that if E_1,E_2,E_3,... are mutually disjoint measurable sets on the line then for any set A, m^* \left(\displaystyle\bigcup_{k=1}^{\infty} (A \cap E_k) \right) = \displaystyle\sum_{k=1}^{\infty}...
19. ### Show whether E is a Measurable Set

Let (E_k)_{k=1}^{\infty} be a sequence of measurable sets in \mathbb{R}^n. Let E be a subset of \mathbb{R}^n so that x \in E if and only if x belongs to exactly 2 of the sets E_k. Determine whether E is measurable. If it is, show it. If not, give a counterexample. I suppose it is measurable...
20. ### Measurable Set Question (2)

Let E be a measurable subset of [a,b]. Let \{ I_k \} be a sequence of open intervals in [a,b] such that m(I_k \cap E) \geq \frac{2}{3}m(I_k), k=1,2,3,.... Prove that m((\cup_{k=1}^{\infty} I_k) \cap E) \geq \frac{1}{3}m(\cup_{k=1}^{\infty} I_k). I've managed to prove below: Suppose that...