課程名稱︰實分析一
課程性質︰數學所必修
課程教師︰劉豐哲
開課學院：理學院
開課系所︰數學所
考試日期（年月日）︰104/01/14
考試時限（分鐘）：100
試題 :
1. (10%) Suppose that A is a Borel set in R^n with λ^n(A)>0 and that f is a
nonnegative measurable function on A, Show that
∫ f dλ^n = sup｛∫ g dλ^n :0≦g≦f, g is a Borel measurable simple fn｝
A A
2. (10%) Suppose that μ measures a metric space X. Show that μ is a
Caratheodory outer measure if and only if B(X) ﹤Σ^μ
3. (20%) Suppose that ω is a nonnegative measurable function on R such that
g(x) = ∫ ω dλ^n < ∞ for all x belongs to R. Show that
(-∞,x]
μ_g(B) = ∫ ω dλ^n for B belongs to B(R).
B
4. (15%) Let f belongs to L^1(R). Show that
lim ∫ |f(x+y)-f(x)| dx = 2∫ |f(x)| dx
y→∞ R R
5. (15%) Suppose that f: R → R is continuously differentiable. Use Vitali
covering theoremto show that λ(f({x belongs to R:f'(x)=0}))=0.
6. (15%) Show that if f belongs to L^p(R^n), 0<p<∞, then
lim ______1_____ ∫ |f(y)-f(x)|^p dλ^n(y)=0 for a.e. x belongs to R
r→0+ λ^n(B_r(x)) B_r(x)
7. (15%) Suppose that f is locally Lebesque integrable on R. Define
F(x) := ∫ (e^(-x-y)-e^(-2(x-y))) f(y) dλ(y) for x belongs to R.
[0,x]
Show that F"+3F'+2F=f holds true a.e. on R.