Integrability of functions $ f,g \in R([a,b]) $ when $ A \subseteq [a,b] $ is a dense set s.t. $ f|_{A} = g|_{A} $

calculusintegrationreal-analysisriemann-integration

Theorem: Let $ f,g : [a,b] \rightarrow \mathbb{R} $ s.t. $ f,g, \in R([a,b]) $. Let $ A \subseteq [a,b] $ be a dense set. Suppose for all $ x \in A $, $f(x) = g(x)$. Then $ \int_{a}^{b} f = \int_{a}^{b} g $.

Proof from lecture notes:
We'll chose a sequence of partitions $ \{ \prod_n \}_{n=1} $ with $ \lambda(\prod_n) \rightarrow 0 $. For every such partition $ \prod_n $ we'll chose a finite sequence of appropriate points $ \{ t_i^{(n)} \}_{i=1}^{m_n} $ where $ m_n $ is the number of points in partition $ \prod_n $, for every $ n $. Also $ t_i^{(n)} \in A $ for every $ i ,n $.
We have that $\forall n. S(f,\prod_n,\{ t_i^{(n)} \}_{i=1}^{m_n}) = S(g,\prod_n,\{ t_i^{(n)} \}_{i=1}^{m_n} ) $ and that is because $ f|_{A} = g|_{A} $. As $ n \rightarrow \infty $ we'll get $ \int_{a}^{b} f = \int_{a}^{b} g $

Questions:

  1. Can you give an example of a choice of sequence of partitions $ \{ \prod_n \}_{n=1} $ with $ \lambda(\prod_n) \rightarrow 0 $?
  2. Can you give an example of a choice of finite sequence of appropriate points $ \{ t_i^{(n)} \}_{i=1}^{m_n} $ of partition $ \prod_n $, for every $ n $ ( s.t. $ t_i^{(n)} \in A $ for every $ i,n $ ) ?
  3. In the proof , why do I need the condition that $A \subseteq [a,b]$ is a dense set ? I can't quite see how the fact that it's dense is used in the proof.

Notes about notation:

  • $ f \in R([a,b]) $ means $ f$ is Riemann integrable on $[a,b] $ .

  • ( mesh of partition ) $ \lambda(\prod _n) = max_{i=1,…,M_n}|{ \triangle x_i}| $ , where $ x_i \in \prod_n $, for all $ 1 \leq i \leq M_n $, where $ M_n $ is the number of points in the partition.

  • ( Riemann sum ) given $ f:[a,b] \rightarrow \mathbb{R} $ a partition $ \prod $ and finite sequence of approproiate numbers $ \{ t_i \} $, their Riemann sum is defined as: $S\left(f, \Pi,\left\{t_{i}\right\}\right)=\sum_{i} f\left(t_{i}\right) \Delta x_{i}$

  • ( finite sequence of appropriate numbers ) Given a partition $ \prod = ( x_0,…,x_l ) $, we'll say $ \{ t_1,…,t_l \} $ are appropriate points for the partition if it occurs that $t_i \in [ x_{i-1} , x_i] $ for all $ i = 1,…,n $

  • ( Reminder for dense sets: ) Let $ I \subset \mathbb{R} $ be an interval ( finite or not ). set $ A \subset I $ will be called dense in $ I $ if for every open interval $ J \subseteq I $ it occurs that $ A \cap J \neq \emptyset $

Thanks in advance for help!

Best Answer

  1. If $[a,b] = [0,1]$ you obviously want to consider $t_i^{(n)} = \frac i n$ and let $i$ going from $0$ to $n$. For general $a$ and $b$ just take $t_i^{(n)} = a + i \frac {b-a}{n}$.
  2. Take any sequence of positive real numbers $\lambda_n$ that vanishes at $+ \infty$. Take $(t_i^{(n)})$ the mesh defined in point 1. For any $i,n$, by density of $A$, $$ A \cap ] t_i^n - \lambda_n , t_i^n + \lambda_n [ \cap [a,b]\neq \emptyset.$$ Then call $x_i^n$ an element of this set, $$ |x_{i+1}^n - x_i^n| \leq |x_{i+1}^n - t_{i+1}^n| + |t_{i+1}^n - t_i^n| + |t_i^n - x_i^n| < 2 * \lambda_n + \frac 1 n $$ so the mesh $(x_i^{(n)})$ mesh goes to $0$.
  3. As you saw in point 2. we need density. Without it just take $A = \emptyset$. To see why it works, prove that if $f,g$ are continuous and equal on $A$ then $f=g$.
Related Question