1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an example of an unbounded open set with finite measure. Justify your answer, 3) If a is a single point on the number line show that m ( a ) = O. 4) Prove that if K is compact and U is open with K U then m(K) m(U). 5) show that the Cantor set C is compact and m(C) = O. 6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral of the following functions a) f : [O, 1] IR defined by f(x) = O if x is rational and f(x) = 2 if x is irrational. b) f : [O, 1] --- IR defined by f(x) = x2 for all x E [O, 1]
1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an exa...
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral...
Please prove Problem 11 & 12 carefully (note that m represents Lebesgue measure & m* represents Lebesgue outer measure): 11. Let E c Rn be an arbitrary subset. Show that for all є > 0 there exists an open set G containing E with m(G) m"(E) +e. 12. Let E C Rn be a measurable subset. Show that for all € > 0 there exists an open set G containing Ewith m (G\ E) < є. 11. Let E c...
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a measurable subset of U, then the area of F(A) is given by area(F(A)) - A det Jrx, )dm(x, y) If F is the induced map of a holomorphic f, what is the resulting formula? 2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number α ∈ [0, 1] (as a subset of the reals now!) and for each (rational) q ∈ [0, 1] look for an...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
1.) Use the definition of the outer measure to show that m*((0,2)) = 2 2.) Let E be a lebesgue measurable set, show that E compliment is also lebesgue measurable?
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] n Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number a € [0, 1] (as a subset of the reals now!) and for each (rational) qe [0, 1] look for an open...