Show that limn→∞∫
1
0
g(x)f(nx)dx=(∫
1
0
g(x)dx)(∫
1
0
f(x)dx)I am new to this type of mathjax formatting, so feel free to refine this question. I've tried my best with the formatting.
Anyway, this is a question I've encountered in a homework sheet, and I do not even know where to begin:
Let f:[0,∞)→R
be a continuous real valued function such that f(x+1)=f(x)
for all x≥0
. If g:[0,1]→R
is an arbitrary continuous function, show that
lim
n→∞
∫
1
0
g(x)f(nx)dx=(∫
1
0
g(x)dx)(∫
1
0
f(x)dx).
We were given a hint:
∫
1
0
g(x)f(nx)dx=
1
n
n
∑
i=1
∫
i
i−1
g(
u
n
)f(u)du,
and put t=u−i+1.
I have absolutely no idea where to begin. My thoughts are that this question will involve the use of Lebesgue's convergence theorem, and perhaps the monotone convergence theorems. I have a very basic understanding of these theorems but still struggle when they need to be applied. My understanding is that:
Function must be Riemann Integrable
fn→f
almost everywhere
|fn|≤g∈L1
I understand 1., and kind of understand 3. but I'm never able to prove 2. without any help. In fact, I only have a vague understanding of 2. and 3.
I've attempted many questions, but just cannot finish one without help. My take is that I lack basic understanding on measure theory and I also lack practice, although I've been spending a large portion of time on this subject this semester. Everything is new and extremely difficult. I'd appreciate some books on the subjects, the lecture notes are really good, but I don't think it's enough at this point. I want something better than a mere pass (and if I aim for a pass, I will fail the subject) -- I want to actually understand it.
Any help and recommendations are appreciated. Thanks!
0
g(x)f(nx)dx=(∫
1
0
g(x)dx)(∫
1
0
f(x)dx)I am new to this type of mathjax formatting, so feel free to refine this question. I've tried my best with the formatting.
Anyway, this is a question I've encountered in a homework sheet, and I do not even know where to begin:
Let f:[0,∞)→R
be a continuous real valued function such that f(x+1)=f(x)
for all x≥0
. If g:[0,1]→R
is an arbitrary continuous function, show that
lim
n→∞
∫
1
0
g(x)f(nx)dx=(∫
1
0
g(x)dx)(∫
1
0
f(x)dx).
We were given a hint:
∫
1
0
g(x)f(nx)dx=
1
n
n
∑
i=1
∫
i
i−1
g(
u
n
)f(u)du,
and put t=u−i+1.
I have absolutely no idea where to begin. My thoughts are that this question will involve the use of Lebesgue's convergence theorem, and perhaps the monotone convergence theorems. I have a very basic understanding of these theorems but still struggle when they need to be applied. My understanding is that:
Function must be Riemann Integrable
fn→f
almost everywhere
|fn|≤g∈L1
I understand 1., and kind of understand 3. but I'm never able to prove 2. without any help. In fact, I only have a vague understanding of 2. and 3.
I've attempted many questions, but just cannot finish one without help. My take is that I lack basic understanding on measure theory and I also lack practice, although I've been spending a large portion of time on this subject this semester. Everything is new and extremely difficult. I'd appreciate some books on the subjects, the lecture notes are really good, but I don't think it's enough at this point. I want something better than a mere pass (and if I aim for a pass, I will fail the subject) -- I want to actually understand it.
Any help and recommendations are appreciated. Thanks!
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