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--- math104-s21:hw12 [2021/04/23 21:27]
pzhou created
+++ math104-s21:hw12 [2022/01/11 10:57] (current)
pzhou ↷ Page moved from math104-2021sp:hw12 to math104-s21:hw12
@@ Line 1: / Line 1: @@
 ====== HW 12 ======
-Problem 1 and 2 involves Riemann integral $\int f dx$ instead of the more general Riemann Stieltjes integral.
+. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \subset [a,b]$, $f$ is integrable on $[c,d]$.
-. (2 point) Show that if $f$ is integrable on $[a,b]$, then for any sub-interval $[c,d] \In [a,b]$, $f$ is integrable on $[c,d]$.
 . (2 point) If $f$ is a continuous non-negative function on $[a,b]$, and $\int_a^b f dx = 0$, then $f(x)=0$ for all $x \in [a,b]$.
@@ Line 9: / Line 7: @@
 . (3 point) Let $f:[0,1] \to \R$ be given by
 $$ f(x) = \begin{cases}
-&\text{if } x = 0 \\
+&\text{if } x = 0 \cr
-   \sin(1/x) &\text{if } x \in (0,1]
+\sin(1/x) &\text{if } x \in (0,1]
 \end{cases}.
 $$
 And let $\alpha: [0, 1] \to \R$ be given by
 $$ \alpha(x) = \begin{cases}
-&\text{if } x = 0 \\
+&\text{if } x = 0 \cr
-   \sum_{n \in \N, 1/n<x} 1/2^n &\text{if } x \in (0,1]
+\sum_{n \in \N, 1/n<x} 2^{-n} &\text{if } x \in (0,1]
 \end{cases}.
 $$
-Is $f$ integrable with respect to $\alpha$ on $[0,1]$?
+Prove that $f$ is integrable with respect to $\alpha$ on $[0,1]$.
+Hint: prove that $\alpha(x)$ is continuous at $x=0$.
+. (3 point) Let $p,q>0$ be positive real numbers, such that $1/p + 1/q = 1$. Prove that,
+if $f, g$ are bounded real functions on $[a,b]$ that are Riemann integrable, then
+$$ \int fg dx \leq \left[ \int |f|^p dx \right]^{1/p} \left[ \int |g|^q dx \right]^{1/q} $$
+Hint:
+(a) If $u \geq 0, v \geq 0$, then $$ uv \leq \frac{u^p}{p} + \frac{v^q}{q} $$
+If you cannot prove this, you may assume it and proceed (no points taken off). If you want to prove it, you may fix $u$ and let $v$ vary from $0$ to $\infty$, and watch how $\frac{u^p}{p} + \frac{v^q}{q} - uv$ change, and obtain that at the minimum the quantity is still non-negative.
+(b) If $f, g$ are **non-negative** Riemann integrable functions on $[a,b]$, and
+$$ \int f^p dx = 1, \quad \int g^q(x) dx = 1 $$
+Show that $\int fg dx \leq 1$.
+Suggested reading: \\
+. Ross theorem 32.7, if a function $f$ is Riemann integrable on $[a,b]$, then as 'mesh-size' of a partition goes to 0, the gap $U(P, f) - L(P, f)$ tends to 0.
+. There is a 'Lebesgue criterion for Riemann integrability', see [[http://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/Lebesgue%20Criterion%20for%20Riemann%20Integrability.pdf | here]].   A weaker version that avoids introducing Lebesgue measure is the following:  if $f:[a,b] \to \R$ is bounded and real, and $f$ has **countably many** discontinuities, then $f$ is Riemann integrable. You can try to prove this using a similar strategy to Theorem 6.10 in Rudin.
-. (3 point)

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