# Search results

1. ### Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

I like the sandwhich approach better than mine. Thanks for the pointers. I really lost the forest from the trees on this problem. I got it stuck in my head that perhaps the irrationals were "dense enough" to create 2 subsets with the same measure. It didn't dawn on me to use L.measure...
2. ### Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

Yep. I messed that up too. One more time. By DeMorgan, C\left(\cap_{k=1}^{\infty} E_k\right)=C\left[\left(\bigcup_{k=1}^{\infty}E^{c}_{k}\right)^{c}\right]=\bigcup_{k=1}^{\infty}E^{c}_{k} Consider: m\left(\bigcup_{k=1}^{\infty}E^{c}_{k}\right)...
3. ### Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

By DeMorgan, C\left(\cap_{k=1}^{\infty} E_k\right)=\left(\bigcup_{k=1}^{\infty}E^{c}_{k}\right)^{c} A lower bound immediately comes to mind. I'm still stewing on how to get an upper bound to pop out. Lower Bound: Consider: m\left[\left(\bigcup_{k=1}^{\infty}E^{c}_{k}\right)^{c}\right]...
4. ### Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

I answered your question the best I could. Your question is closely related to the question I am posing. If I could simply answer it, then I wouldn't've started the thread.
5. ### Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

If I can construct a counterexample of 2 disjoint and noncountable subsets on [0,1] of measure 1, then the measure of the complement of intersections would be 1. However, I have not been able to find such a counterexample. The answer based on what I have considered is 0.
6. ### Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

Homework Statement Let \left\{E_{k}\right\}_{k\in N} be a sequence of measurable subsets of [0,1] satisfying m\left(E_{k}\right)=1. Then m\left(\bigcap^{\infty}_{k=1}E_{k}\right)=1. Homework Equations m denotes the Lebesgue measure. "Measurable" is short for Lebesgue-measurable. The Attempt...
7. ### Estimate integral. (Lp Spaces, Holder)

I was hoping that I was missing a more strategic approach. (I had already tried working through it with q=3 but ran into unboundedness of g. Working with 2 solved the unboundedness problem, but missed the mark.) Problem does solve with p=3 and q=3/2. Thanks for the help.
8. ### Estimate integral. (Lp Spaces, Holder)

Homework Statement Show that: \left(\int^{0}_{1}\frac{x^{\frac{1}{2}}dx}{(1-x)^{\frac{1}{3}}}\right)^{3}\leq\frac{8}{5} Homework Equations Holder inequality. The Attempt at a Solution First, I took the cube root of each side. This let me just deal with the 1-norm on the left. Then I...
9. ### Proof: Integral is finite (Fubini/Tonelli?)

Here's what I am thinking. Consider: \int_{[0,1]}f(y)\left[\int_{[0,1]}\frac{1}{|x-y|^{1/2}}dx\right]dy=\int_{[0,1]}f(y)\left[2\left(\sqrt{1-y}-\sqrt{y}\right)\right]dy\leq\int_{[0,1]}f(y)\cdot 2<∞. Therefore \int_{[0,1]^{2}}\frac{f(y)}{|x-y|^{1/2}}<\infty by Tonelli's Theorem. Then...
10. ### Proof: Integral is finite (Fubini/Tonelli?)

Homework Statement Let f:[0,1]→ℝ be non-negative and integrable. Prove that \int_{[0,1]}\frac{f(y)}{|x-y|^{1/2}}dy is finite for ae x in [0,1] Homework Equations This looks like a Fubini/Tonelli's Theorem problem from the problem givens. The Attempt at a Solution I honestly don't know...
11. ### Sigma Algebra

The definition the text gave for an Rn interval was the cross product of: av ≤ xv ≤ bv (v= 1, 2, ..., n). It acknowledged other intervals (open, semi open), but stated that intervals should be assumed to be closed unless specifically mentioned.
12. ### Sigma Algebra

Yeah. I noticed those problems when I went back to the problem (again). Here's where I am with the problem: Let I be the collection of all intervals of Rn and ƩI be the σ-algebra generated by I. Let G be the set of all open subsets of Rn and ƩG be the σ-algebra generated by G. Pick an...
13. ### Sigma Algebra

I got confirmation from the professor that he uses the word "coincide" to mean "equal". So both directions of containment need to be proved. I am also changing the notation from Ʃ(x) to Ʃx to avoid confusion that Ʃ is a function. This was my first attempt to show \Sigma_{G}\subset\Sigma_{I}...
14. ### Are Sigma Algebras Unique for a Given Set?

Thanks for confirming this. I spoke to the professor about this. His comment was "My native language is Russian. Questions clarifying how I use articles are completely fair."
15. ### Are Sigma Algebras Unique for a Given Set?

Is there only 1 σ-algebra generated for a set? Consider M={1,2}. Ʃ(M)={∅,M} satisfies the definition of a σ-algebra. However, Ʃ(M)={∅,{1},{2},{1,2}} also satisfies the definition of a σ-algebra. However, the way that my text presents these problems (Prove that the σ-algebra generated...)...
16. ### Sigma Algebra

Thanks for the help Bacle2. Here is what I am turning in. Proof: Let I be the collection of all intervals of Rn, and Ʃ(I) be the σ-algebra generated by I. Let O be the set of all open subsets of Rn, and Ʃ(O) be the σ-algebra generated by O. To show: Ʃ(I) is a subset of Ʃ(O). Pick an...
17. ### Sigma Algebra

Homework Statement Prove that the σ-algebra generated by the collection of all intervals in Rn coincides with the σ-algebra generated by the collection of all open subsets of Rn. Homework Equations A σ-algebra is a nonempty collection Σ of subsets of X (including X itself) that is closed...
18. ### Laurent Series Expansion Centered on z=1

I think I figured the other fraction out: \frac{1}{2}<|z−1|<2,\frac{1}{2(z-1)}<1. Hence...
19. ### Laurent Series Expansion Centered on z=1

No biggie. So is the last expression the answer then?
20. ### Laurent Series Expansion Centered on z=1

You're right about the -1. It should be raise to n, not n+1. I mistakingly thought the index started at 1 instead of 0. I'm pretty sure the coefficient of the (z-1) term is a multiple of 2 in this case.
21. ### Laurent Series Expansion Centered on z=1

It's this, right? \frac{1}{2z-1}=\sum^{\infty}_{n=0}(-1)^{n+1}(z-1)^{n}*2^{n} So the expansion would become: f(z)=-\frac{1}{5}\sum^{\infty}_{n=0}(z-1)^{n}[(-2)^{n+1}-2^{-n-1}]
22. ### Laurent Series Expansion Centered on z=1

Homework Statement Find a Laurent Series of f(z)=\frac{1}{(2z-1)(z-3)} about the point z=1 in the annular domain \frac{1}{2}<|z-1|<2. Homework Equations The Attempt at a Solution By partial fraction decomposition...
23. ### Maximum Value Complex Function

Thanks for the help, Dick. I really appreciate it.
24. ### Maximum Value Complex Function

How does this look? Let |z|≤1 be a domain, D. Let f(z)=(z-1)(z+1/2). Observe f(z) is the product of 2 analytic functions: g(z)=z-1 and h(z)=z+1/2. Therefore f(z) is analytic on D. Since f(z) is analytic on D, it is also continuous on D. By Maximum Modulus Theorem, the max|f(z)| occurs on the...
25. ### Maximum Value Complex Function

Homework Statement Find the maximum value of |(z-1)(z+1/2)| for |z|≤1. Homework Equations Calculus min/max concepts? The Attempt at a Solution Let f(z)=|(z-1)(z+1/2)|. Observe f(z) is the product of 2 analytic functions on |z|≤1, g(z)=z-1 and h(z)=z+1/2. Therefore f(z) is analytic on...
26. ### Non-rational Exponents?

General Question: I don't have a good grasp of what it means to use a non-rational exponent, such as xe or xi. For irrational powers, the best I have is to look at xe (for example) as a series xn where n converges to e using rational n's. That's an analytical approach that feels right, but...
27. ### Epsilon-Delta Proof of Limit in R2

Homework Statement Examine the limit behavior on (ℝ2, 2-norm) for f(λ,μ)=\frac{λμ}{|λ|+|μ|} defined on ℝ2\{(0,0)} at (λ,μ)=(0,0). I think the limit of f(λ,μ) at (0,0) is (0,0). I have tried multiple paths to (0,0) that seem to indicate this. (λ=μ, λ=μ2, et al.) However, I am lost trying...
28. ### Banach Fixed Point and Differential Equations

\int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\int^{x}_{0}(1-\frac{1}{1+t^{2}})dt Let t=tanθ. Then dt=sec2θdθ. \int^{x}_{0}(1-\frac{1}{1+t^{2}})dt=\int^{x}_{0}dt-\int^{x}_{0}\frac{1}{sec^{2}\vartheta}sec2θdθ=\int^{x}_{0}dt-\int^{x}_{0}dθ=t-arctan(t) from 0 to x=x-arctan(x). x-arctan(x)=.5 at...
29. ### Banach Fixed Point and Differential Equations

Homework Statement Find the value of x, correct to three decimal places for which: \int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}. Homework Equations Banach's Fixed Point Theorem Picard's Theorem? The Attempt at a Solution I'm not sure where to start with this type of problem...
30. ### Deduce that (P[0,1], norm(inf)) is not complete.

The remainder term of a Taylor series would go to zero as the number of terms approaches infinity?