課程名稱︰實分析一
課程性質︰數學研究所必選修、應用數學科學研究所必選修、數學系選修
課程教師︰劉豐哲
開課學院：理學院
開課系所︰數學系
考試日期︰2014年01月
考試時限：110分鐘
試題 :
Real Analysis I (Fall 2013)
Final Examination
1. (20%) A family {f_α} of integrable function on a measure space (Ω,Σ,μ)
is called uniformly integrable if for any ε > 0, there is δ > 0 such that
if A is contained by Σ with μ(A) ≦ δ, then ∫|f_α|dμ≦ε for all α.
A
Show that if {f_n} is uniformly integrable sequence of functions on Ω which
converges a.e. to an integrable function f on Ω, then
lim ∫|f_n - f|dμ = 0.
n->∞
n
2. Let ω≧0 be integrable on R and let μ be a premeasure defined for open
sets G in R^n by n
μ(G) = ∫ωdλ
n G
Denote by μ* the measure on R constructed from μ by Method I.
(a) (6%) Show that μ*(S) = inf μ(G) where infimum is taken over all open
sets G containing S.
(b) (7%) Show that μ* is a Caratheodory measure and
n
μ*(B) = ∫ωdλ
B
for Borel sets B. μ* n
(c) (7%) Show that L^n is contained by Σ and μ*(A) = ∫ωdλ if A
n A
belongs to L .
3. (20%) Define a function f on (0,∞) by
∞ e^(-xt^2)
f(x) = ∫ -----------dt, x belongs to (0,∞).
0 1 + t^2
Show that f is continuously differentiable on (0,∞) and is a solution of
√π 1
the equation y' - y + ----- ------ = 0 on (0,∞).
2 √x
4. (a) (5%) A function f on [a,b] is called Lipschitz if there is L > 0 such
that |f(x) - f(y)|≦L|x-y| for all x,y in [a,b]. Show that a Lipschitiz
function is AC.
(b) (15%) Let f be a continuous BV function on [a,b]. Show that f is AC if
and only if there exists a sequence {f_n} of Lipschitz functions such
b
that lim V (f - f_n) = 0.
n->∞ a
5. (a) (10%) Let f be an integrable function on [a,b] with the property that
b
∫fg'dλ = 0.
a
for all AC functions g such that g(a) = g(b) = 0. Show that f = constant
a.e.
(b) (10%) Let f and g be integrable functions on [a,b] and suppose that
b b
∫fh'dλ = -∫ghdλ
a a
for all AC functions h with h(a) = h(b) = 0. Show that f is equivalent
^ ^
to an AC function f and f' = g a.e.
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