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...
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)...
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]...
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.
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.
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...
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.
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...
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...
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...
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.
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...
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}...
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."
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...)...
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...
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...
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.
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}]
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...
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...
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...
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...
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...
\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...
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...