11 > 0, then there exists a subset of A that is not Prove that if...
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...
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
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...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
equivalent 4. Let E C R. Prove that the following statements are (a) E is Lebesgue measurable (b) Given e> 0, there exist m* denotes the Lebesgue measure of a set (c) Given e 0, there exist a closed set F such that F C E and m* (E- F) < E. (d) There exists a set G (a countable intersection of open sets) such that E C G and m* (G - E) 0 (e) There exists a set...
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.
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
A metric space (X, d) is totally bounded if, given
ε>0, there exists a finite subset =
of X, called an ε-net, such that for each x∈X there
exists
∈
such that d(x,)
< ε. Prove that if Y is a subset of a totally bounded space X
then, given ε>0, the subset Y has an ε-net and
therefore Y is also totally bounded.
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