Analysis III TT23, Basic theorems about the integral
Flashcards
When proving that if $f$ and $g$ are integrable on $[a, b]$ then so are $\max(f, g)$, $\min(f, g)$ and $ \vert f \vert $, 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 $ \vert f \vert $ in a way that makes this easier?
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
\[ \vert f(x) - f(y) \vert \le \vert x - y \vert \]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 $\int^b _ a f$?
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 $ \vert \int^b _ a f \vert \le \int^b _ a \vert f \vert $?
Consider $f, \vert f \vert $ and $-f, \vert f \vert $.
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?
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).
@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.
@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).
@Prove that if $f$ and $g$ are integrable on $[a, b]$ then so are $\max(f, g)$, $\min(f, g)$ and $ \vert f \vert $
Todo (analysis iii, page 9).
@Prove that if $f$ and $g$ are integrable on $[a, b]$ then so is $fg$.
Todo (analysis iii, page 10)
@Prove that integration preserves weak inequalities.
Todo (analysis iii, page 9).
@Prove that any continuous function $f: [a, b] \to \mathbb R$ is integrable.
Todo (analysis iii, page 13)
@Prove that if the integral of a continuous function $f: [a, b] \to \mathbb R$ is zero, then that function is zero everywhere.
Todo.