# Notes - Analysis III TT23, Basic theorems about the integral

> Source: https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/basic-theorems-about-the-integral/ · Updated: 2023-04-26 · Tags: uni, notes

- [Course - Analysis III TT23](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/)

### Flashcards
When proving that if $f$ and $g$ are integrable on $[a, b]$ then so are $\max(f, g)$, $\min(f, g)$ and $|f|$, you can save yourself a lot of work and actually only prove one statement about $f$ and then just use linearity. How can you rewrite $\max(f, g)$, $\min(f, g)$ and $|f|$ in a way that makes this easier?::
$$
\begin{aligned}
\max(f, g) &= g + \max(f - g, 0) \\
\min(f, g) &= -\max(-f, -g) \\
|f| &= \max(f, -f)
\end{aligned}
$$
so only need to prove the $\max(h, 0)$ case.

@Define a “non-expanding function” (i.e. a metric map).::
A function $f$ such that
$$
|f(x) - f(y)| \le |x - y|
$$

Suppose $x \le x’$ and $y \le y’$ and that
$$
x+y = x' + y'
$$
Then what is true?::
- $x = x’$
- $y = y’$

Suppose $f$ is integrable on $[a, b]$. What easy bound can you put on the value of $\int^b_a f$?::
$$
(b - a)\inf f \le \int_a^b f \le (b-a)\sup f
$$

Suppose $f$ is integrable on $[a, b]$. Then 
$$
(b - a)\inf f \le \int_a^b f \le (b-a)\sup f
$$
How can you use this to prove that if $f \le g$ pointwise, $\int^b_a f \le \int^b_a g$?::
Apply the above to $g - f$.

Suppose $f, g$ are integrable on $[a, b]$ with $f \le g$ pointwise. Then
$$
\int^b_a f \le \int^b_a g
$$
How can you use this to prove that $|\int^b_a f| \le \int^b_a |f|$?::
Consider $f, |f|$ and $-f, |f|$.

When proving that the product of two integrable functions $f$ and $g$ is integrable, how can you rewrite $f$ and $g$ in order to make it so you only need to prove the statement for non-negative functions?::
Let $f = f_+ - f_-$ where $f_+ = \max(f, 0)$ and $f_- = -\min(f, 0)$ and likewise for $g$. Then
$$
fg = f_+g_+ - f_+g_- - f_-g_+ + f_-g_-
$$

When proving that the product of two integrable functions $f$ and $g$ is integrable, you get to a point where you show that the minorants $\phi_-, \psi_-$ and majorants $\phi_+, \psi_+$ naturally give a minorant $\phi_-\psi_-$ and majorant $\phi_+ \psi_+$ for $fg$, but still need to show that $I(\phi_+ \psi_+) - I(\phi_-\psi_-) < \varepsilon$. Assuming $\phi_+, \phi_-, \psi_-, \psi_+ \le M$ for some $M$, how can you rearrange
$$
I(\phi_-\psi_-) - I(\phi_+ \psi_+) < \varepsilon
$$
to show it can be made arbitrarily small?::
$$
\begin{aligned}
I(\phi_+ \psi_+) - I(\phi_-\psi_-) &= I(\phi_+ \psi_+) -I(\phi_+ \psi_-) + I(\phi_+ \psi_-) -  I(\phi_-\psi_-) \\
&= I(\phi_+)(I(\psi_+) - I(\psi_-)) + I(\psi_+)(I(\phi_+) - I(\phi_-)) \\
&< M\varepsilon + M\varepsilon \\
&= 2M\varepsilon
\end{aligned}
$$

When proving that a continuous function $f: [a, b] \to \mathbb R$ is integrable, what key fact lets you do this?::
Any continuous function on a closed interval is uniformly continuous.

### Proofs
@Prove that if $f$ is integrable on $[a, b]$, then for any $a < c < b$, $f$ is Riemann integrable on $[a, c]$ and on $[c, b]$ and that $\int^b_a f = \int^c_a f + \int^b_c f$.::
Todo (analysis iii, page 8).

@todo~

@Prove that if $f$ and $g$ are integrable functions on $[a, b]$, then $\lambda f + \mu g$ is an integrable function on $[a, b]$ and its integral is given by
$$
\lambda \int^b_a f + \mu \int^b_a g
$$

::

Todo.

@todo~

@Prove that if $f$ is integrable on $[a, b]$ and $\hat f$ differs from $f$ at finitely many points, then $\hat f$ is also integrable.::
Todo (analysis iii, page 9).

@todo~

@Prove that if $f$ and $g$ are integrable on $[a, b]$ then so are $\max(f, g)$, $\min(f, g)$ and $|f|$::
Todo (analysis iii, page 9).

@todo~

@Prove that if $f$ and $g$ are integrable on $[a, b]$ then so is $fg$.::
Todo (analysis iii, page 10)

@todo~

@Prove that integration preserves weak inequalities.::
Todo (analysis iii, page 9).

@todo~

@Prove that any continuous function $f: [a, b] \to \mathbb R$ is integrable.::
Todo (analysis iii, page 13)

@todo~

@Prove that if the integral of a continuous function $f: [a, b] \to \mathbb R$ is zero, then that function is zero everywhere.

::

Todo.

@todo~

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt