Entrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim


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1 1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your computtion: lim e x2n dx. 2. Let f be rel function defined on [, 1] with f() = 1. If the set A = {x : f(x) > } is both open nd close in [,1], then is there δ > such tht f(x) δ on [, 1]? Prove your conclusion. 3. Let f be rel function defined on [, 1]. If f is monotone incresing, nd be the set of points on [, 1] where f is discontinuous. Show tht m =. 4. Let f L(, ), show tht lim λ f(x) cos(λx)dx =. 5. Let {f n } be sequence of mesurble functions on (, b), nd Show tht is mesurble. = {x (, b) : f n (x) is convergent}. 6. Let f, g be two bsolutely continuous functions on [, 1]. Prove tht fg is bsolutely continuous on [, 1]. Is it lso true if the intervl [, 1] is replced by (, )? Prove your conclusion. 7. Let {f n } n be sequence of integrble functions on [, 1] such tht f n (x) x.e. on [, 1] nd f n (x)dx 1 2 Does f n (x) converge to x in L 1 [, 1]? Prove your conclusion. 8. Let f L (, 1). Prove lim p f p = f. 1
2 Attempt to solve ll 8 problems Ph.D. ntrnce xm Rel Anlysis April 18, Let {f n } be sequence of continuous functions which converges uniformly to f on set R. Prove: for every sequence {x n } in tht is convergent to x, there holds lim f n (x n ) = f(x). 2. Let f : R R be continuous function, nd {K n } be decresing sequence of compct subsets of R. Show tht f( K n ) = f(k n ). 3. Let m denote the Lebesque mesure on R nd let R be Lebesque mesurble subset. Suppose < α < m(). Show tht there is compct subset K of R such tht K nd m(k) = α. 4. () Prove: If f is mesurble in, then for every α R, the set {x : f(x) = α} is mesurble. (b) Give n exmple to show tht the converse of () is not true. 5. Let f be nonnegtive integrble function on. Prove: for ny ɛ >, there exists δ > such tht for every subset A with m(a) < δ, there holds f < ɛ. 6. Let {f n } be sequence of integrble functions on [, 1] such tht f n f.e. on [, 1] with f integrble. Prove: lim f n f = if nd only if lim A f n = 7. Let f be nonnegtive integrble function on mesurble set with m() >. () Prove: If f =, then f =.e. on. (b) Prove: If f n = f > for ll n = 1, 2,, then f = χ F.e. on for some mesurble set F with m(f ) > (Here χ F denotes the chrcteristic function of the subset F ). 8. Let {f n } be sequence in L p [, b] for some p > 1, nd let q stisfy 1/p + 1/q = 1. () If f n is convergent in L p [, b], is it true tht (b) If g L q [, b]? Prove your conclusion. f n g f n g f. is convergent for ny is convergent for every g L q [, b], is it true tht f n is convergent in L p [, b]? Prove your conclusion.
3 Choose six of the following: 1. For bounded set, define Ph.D. ntrnce xm Rel Anlysis April 25 m () = b m ([, b] \ ), where [, b] is n intervl contining, nd m denotes the usul outer mesure. Prove the following sttements. () If be the set of ll irrtionl numbers in [, 1], then m () = 1. (b) m () is independent of the choice of [, b], s long s it contins. (c) m () m (). 2. Let be mesurble set in [, 1] with m = c ( 1 2 < c < 1). Let 1 = + = {x+y; x, y }. Show tht there exists mesurble set 2 1 such tht m 2 = Let f(x) be monotone incresing on [, 1] with f() = nd f(1) = 1. If the set {f(x); x [, 1]} is dense in [, 1], show tht f is continuous function on [, 1]. Is it bsolutely continuous on [, 1]? Prove your conclusion. 4. Let f n (x) be sequence of continuous functions on [,1] nd f n (x) f n+1 (x) (n = 1, 2, ). For every x [, 1], lim f n (x) <. Determine nd prove if there is δ > such tht 5. Let f L 1 (R 1 ) nd define lim f n(x) δ x [, 1]. F (t) f(x) sin(xt)dx. Prove F (t) is continuous in R. Is F (t) uniformly continuous in R? Prove your conclusion. 6. Let {f n } be sequence of mesurble functions on (, b), nd Show tht is mesurble. 7. () Stte Ftou s Lemm. = {x (, b) : f n (x) is convergent}. (b) Show by n exmple tht the strict inequlity in Ftou s Lemm is possible. (c) Show tht Ftou s Lemm cn be derived from the Monotone Convergence Theorem. 8. Suppose f is nonnegtive integrble function on [, 1]. If f n = f for ll n = 1, 2,, then f(x) must be the chrcteristic function of some mesurble set [, 1].
4 Ph.D. ntrnce xm Rel Anlysis August 3, 24 (Choose exctly six of the following eight problems.) 1. Use the definition of Lebesgue mesure show tht the set of ll rtionl numbers in [,1] is Lebesgue mesurble set. 2. Let f, g be two bsolutely continuous functions on [, 1]. Prove tht fg is bsolutely continuous on [, 1]. Is it lso true if the intervl [, 1] is replced by (, )? Justify your conclusion. 3. If f(x) is integrble over [, 1], then lim λ f(x)cos(λx)dx =. 4. Let f be function in (,1) defined by setting if x irrtionl f(x) = sin( 1 q ) if x = p q in lowest terms. Find the set C of points where f is continuous nd the set D of points where f is discontinuous. Justify your conclusion. Is function f Riemnn integrble on (,1)? Lebesgue integrble on (,1)? Justify your conclusion. 5. Let [, 1] be closed nd with no interior point. Is it true tht the mesure m() =? Justify your conclusion. 6. Let f L 1 (R 1 ) nd g L (R 1 ), nd define F (t) g(x)f(x + t)dx. Prove F (t) is continuous. Is F (t) uniformly continuous on R 1? Justify your conclusion. 7. Let f(x) = x cos π x for < x 1 nd f() =. () Is f continuous on [, 1]? (b) Is f uniformly continuous on [, 1]? (c) Is f bsolutely continuous on [, 1]? Justify your conclusion. 8. Let {f n } be sequence of rel Lebesgue mesurble functions on [, 1]. If for ny rel g(x) L 2 [, 1], the sequence of rel numbers g(x)f n (x)dx converges. Does f n (x) converge to function f(x) in L 2 [, 1]? Justify your conclusion.
5 NAM (print): Anlysis Ph.D. ntrnce xm, August 29, 23 Solve five exercises from the following list. Write solution of ech exercise on seprte pge. This is two hours exm. In wht follows R stnds for the rel line nd m for the Lebesgue mesure. x. 1. Let 1 2 be n infinite sequence of mesurble subset of R nd ssume tht n =. () Show tht if m( 1 ) < then lim m( n )=. (b) Give n exmple showing tht the conclusion of () my be flse when m( 1 )=. x. 2. Let f n :(, 1] [, ) be decresing sequence of continuous functions converging pointwise to zero function θ. Must f n converge uniformly? x. 3. Is the product of two integrble functions from R to R integrble? Prove it or give counterexmple. x. 4. Show tht exists nd find its vlue. x n +1 lim x n +2 x. 5. Let {f n } be sequence of mesurble functions on [, 1]. Describe the three concepts of convergence stted in (i) (iii) nd give ny implictions between them. The implictions must be proved. (Sketch is enough.) The lck of ech impliction must be supported by counter exmple. (i) f n in mesure s n (ii) f n.e. s n (iii) f n 1 s. x. 6. Suppose tht f is continuous function on [, 1] for which f(t)t n dt =, n =, 1, 2, 3,... [,1] Show tht f is the zero function.
6 Complete ll problems: Ph.D. ntrnce xm Rel Anlysis August 28, Let be set in R with m = β >. Show tht for ny α (, β), there exists set α with m α = α. 2. Suppose 1, 2,, n re n mesurble sets in [, 1], nd every x [, 1] belongs to t lest q of these sets. Show tht, there is t lest n k such tht m k q/n. 3. Let f be nonnegtive nd mesurble on, nd n = {x : f(x) n}. Show tht if n m n <, then f is integrble on, but the converse is not true. 4. Let f(x, y) be bounded function on the unit squre Q = (, 1) (, 1). Suppose for ech y, tht f is mesurble function of x. For ech (x, y) Q, let the prtil derivtive f x 5. Prove: exist. Under the ssumption tht f x d 1 f(x, y) dx = dy is bounded in Q, prove tht f (x, y) dx. y () If f is bsolutely continuous on [, b], then for ny set [, b] with m =, there holds m(f()) =. (b) For continuous nd incresing function f on [, b], if m(f()) = for every in [, b] with m =, then f is bsolutely continuous on [, b]. 6. For f L p [, b] (p > 1), set f = outside of [, b] nd define Show tht f h (x) = 1 x+h f(t)dt for h >. 2h x h f h p f p nd lim h + f h f p =. (Note: You cn use the fct, without giving its proof, tht for integrble φ, there holds φ h (x) dx φ(x) dx.)
7 Ph.D. ntrnce xm Rel Anlysis August 25, 2 Complete ll problems. 1. For ech sttement below, either prove it (if true) or give counter exmple (if flse). () is mesurble if nd only if m (P Q) = m (P ) + m (Q) for ny P nd ny Q Ẽ. (b) If is countble set, then is mesurble nd m =. (c) If is mesurble with m =, then is countble set. (d) If f is mesurble, then so is f. (e) If f is mesurble, then so is f. (f) If f is mesurble on nd f =, then f =.e. on. (g) If f is continuous on [, b], then f is of bounded vrition on [, b]. 2. Suppose {f n } is sequence of nonnegtive mesurble functions, nd f n converges.e. on. Show tht f n = f n. 3. Show tht function stisfying Lipschitz condition on [, b] is bsolutely continuous. (Note: A function f is sid to stisfy Lipschitz condition on [, b] if there is constnt M such tht f(x) f(y) M x y for ll x, y in [, b]. ) 4. Suppose {f n } nd f re functions in L p [, 1] (p 1), nd f n f.e. Show tht {f n } converges to f in L p if nd only if f n p f p.
8 Ph.D. ntrnce xm Rel Anlysis August 2, 1997 Instruction: Complete 6 of the following 7 problems. In ll these problems mesurble nd integble re in the sense of Lebesque, m denotes the Lebesque outer mesure, nd m the Lebesque mesure, nd the Lebesque integrl. 1. For set [, b] ( bounded intervl) with m = β >, show tht for ny α (, β), there exists set α with m α = α. 2. For mesurble sets n with lim m n = m <, prove tht there holds lim f = f for every integrble function f on. n 3. () Stte the definition of mesurble function. (b) Use the definition to deduce tht, if f is mesurble on mesurble set, then for every α R, the set α = {x : f(x) = α} is mesurble. (c) Construct function f on = (, 1) to show the converse of (b) is not true. (Note: You my ssume the existence of nonmesurble set S (, 1).) 4. Use the Hölder inequlity to estblish the generlized Hölder inequlity: m m m Let p i > 1 with 1/p i = 1. Then f i 1 f i pi for ny f i L p i (, 1). i=1 (Note: It would be sufficient if you just show the cse m = 3.) i=1 5. () Stte the definition of n bsolutely continuous function on bounded intervl [, b]. (b) Prove tht, if f is bsolutely continuous on [, b], then for ny set [, b] with m =, there holds m(f()) =. 6. Suppose {f n } nd f re mesurble functions nd f n f.e. in with m <. Show tht there exists sequence of mesurble sets { k } k= such tht k =, m =, nd f n f uniformly on ech k for k = 1, 2,. k= 7. Construct closed nowhere dense (i.e., Cntorlike) set K [, 1] with < mk < 1. (Note: A set is sid to be nowhere dense if its closure contins no nonempty open intervl.) i=1
9 Complete ll problems: Ph.D. ntrnce xm Rel Anlysis August 28, Let be set in R with m = β >. Show tht for ny α (, β), there exists set α with m α = α. 2. Suppose 1, 2,, n re n mesurble sets in [, 1], nd every x [, 1] belongs to t lest q of these sets. Show tht, there is t lest n k such tht m k q/n. 3. Let f be nonnegtive nd mesurble on, nd n = {x : f(x) n}. Show tht if n m n <, then f is integrble on, but the converse is not true. 4. Let f(x, y) be bounded function on the unit squre Q = (, 1) (, 1). Suppose for ech y, tht f is mesurble function of x. For ech (x, y) Q, let the prtil derivtive f x 5. Prove: exist. Under the ssumption tht f x d 1 f(x, y) dx = dy is bounded in Q, prove tht f (x, y) dx. y () If f is bsolutely continuous on [, b], then for ny set [, b] with m =, there holds m(f()) =. (b) For continuous nd incresing function f on [, b], if m(f()) = for every in [, b] with m =, then f is bsolutely continuous on [, b]. 6. For f L p [, b] (p > 1), set f = outside of [, b] nd define Show tht f h (x) = 1 x+h f(t)dt for h >. 2h x h f h p f p nd lim h + f h f p =. (Note: You cn use the fct, without giving its proof, tht for integrble φ, there holds φ h (x) dx φ(x) dx.)
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