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math104-s21:hw12 [2021/04/23 21:57]
pzhou 		math104-s21:hw12 [2022/01/11 10:57] (current)
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 ====== HW 12 ====== ====== HW 12 ======
  
-We  +1. (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 $[c,d]$. 
- +
-1. (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. (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 $x \in [a,b]$.  2. (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 $x \in [a,b]$. 
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 4. (3 point) Let $p,q>0$ be positive real numbers, such that $1/p + 1/q = 1$. Prove that,  4. (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 if $f, g$ are bounded real functions on $[a,b]$ that are Riemann integrable, then
-$$ \int fg dx \leq [ \int |f|^p dx ]^{1/p} [ \int |g|^q dx ]^{1/q} $$+$$ \int fg dx \leq \left[ \int |f|^p dx \right]^{1/p} \left[ \int |g|^q dx \right]^{1/q} $$
 Hint:  Hint: 
 (a) If $u \geq 0, v \geq 0$, then $$ uv \leq \frac{u^p}{p} + \frac{v^q}{q} $$ (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  (b) If $f, g$ are **non-negative** Riemann integrable functions on $[a,b]$, and 
 $$ \int f^p dx = 1, \quad \int g^q(x) dx = 1 $$ $$ \int f^p dx = 1, \quad \int g^q(x) dx = 1 $$
 Show that $\int fg dx \leq 1$.  Show that $\int fg dx \leq 1$. 
  
 +Suggested reading: \\
 +1. 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. 
  
 +2. 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. 
  
  
math104-s21/hw12.1619240251.txt.gz · Last modified: 2021/04/23 21:57 by pzhou

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