课程名称︰分析导论优二
课程性质︰数学系大二必修
课程教师︰王振男
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/05/12
考试时限（分钟）：180
试题 :
1. Evaluate the following integrals.
(a) (10%) ∬ [x+y] dA, where Q = [0, 2] ×[0, 2], and [t] is the greatest
Q
integer ≦ t.
(b) (10%) ∬ f(x, y) dA, where Q = [0, 1] ×[0, 1] and
Q
╭ 0, if at least one of x, y is irrational,
f(x, y) = ╯
╰ 1/n, if y is rational and x = m/n,
where m and n are relatively prime integers with n > 0.
2. (15%) Give an example to show that φ(f(x)) may not be measurable if φ and f
are measurable.
n
3. (20%) Let f be an extended real-valued measurable function defined on |R .
Show that there exists a Borel function g such that f = g a.e. Hint: First
consider f ≧ 0 and use the approximation of simple functions. For the
+ -
general case, we write f = f - f .
4. (25%) Show that there exists a measurable set which is not Borel. You could
use the fact that a properly-defined inverse of Cantor function, P: [0, 1] →
[0, 1], is measurable, 1-1, and the values of P lie entirely in the Cantor
set. Also, you may need the fact that if f is a real measurable function on
-1
|R and B is a Borel set, then f (B) is a measurable set.
1 1
5. (20%) This is a simple form of Sard's theorem. Let f: I → |R be a C
1
function, where I is an open set of |R . Denote C = {x∈I: f'(x) = 0}. Show
that f(C) is of measure zero. Hint: Let K be a closed interval contained in
I. consider the function g: K ×K → |R
╭ f(y) - f(x) - f'(x)(y-x)
│ ────────────, for x ≠ y,
g(x, y) = ╯ y - x
│
╰ 0, for x ＝ y.
Then show that g is uniformly continuous. Thus for any ε > 0 there exists
δ > 0 such that |g(x, y)| < ε whenever |x-y| < δ. partition K into
subintervals with norm less then δ. Then consider those subintervals that
have nonempty intersection with C.
注: 第四题原题目叙述并不清晰，这里的题目叙述是修改后的。