Analysis III TT23, Step functions and basic definitions
Flashcards
@Define what it means for $\phi : [a, b] \to \mathbb R$ to be a step function.
There exists a partition $\mathcal P$ such that $\phi$ is constant on each open interval $(x _ {i-1}, x _ i)$.
@Define, in the context of step functions, a partition $\mathcal P$ of $[a, b]$.
A finite sequence
\[a = x _ 0 \le x _ 1 \le \ldots \le x _ n = b\]@Define, in the context of step functions, a refinement of a partition $\mathcal P$ of $[a, b]$.
A partition $\mathcal P '$ with a finite sequence
\[a = x' _ 0 \le x' _ 1 \le \ldots \le x _ {n'}'\]where each element in $\mathcal P$ occurs in $\mathcal P'$.
@Define what it means for a step function $\phi$ to be adapted to a partition $\mathcal P$.
It is constant on each of the intervals in the partition.
Suppose $\mathcal P _ 1$ and $\mathcal P _ 2$ are two partitions. Then how can you construct a common refinement to both of them?
Take the union and then set in increasing order.
What is true about step functions and indicator functions?
Every step function can be written as a weighted sum of indicator functions.
@Define the “integral” $I$ of a step function $\phi$.
where $c _ i$ is the value of $\phi$ on $(x _ {i-1}, x _ i)$.
@Define what it means for a step function $\phi$ to be a minorant ($\phi _ -$) or a majorant ($\phi _ +$).
pointwise.
@Define what it means for a function $f$ to be integrable.
If $f$ is integrable, how is $\int^b _ a f$ “squeezed” between its minorants and majorants?
$f$ is integrable if
\[\sup _ {\phi _ -} I(\phi _ -) = \inf _ {\phi _ +} I(\phi _ +)\]
What’s an equivalent definition in terms of $\varepsilon$?
Can you give an example of a function that is not (Riemann) integrable?
\[f = \pmb 1 _ {\mathbb Q}\]
When proving that this function is not Riemann integrable, what fact about open intervals of real numbers is useful?
Any non-empty open interval of real numbers contains both rational and irrational numbers.
@Define the optimal majorant $\phi _ +^\mathcal{P}$ (given a partition $\mathcal P$).
@State an alternative (but equivalent) characterisation of integration using optimal minorants and majorants $\phi _ +^\mathcal{P}$ and $\phi _ -^\mathcal{P}$.
$f$ is integrable if and only if $\forall \varepsilon > 0$ there exists a partition $\mathcal P$ such that
\[I(\phi _ +^\mathcal P) - I(\phi _ -^\mathcal P) < \varepsilon\]@Prove the forward direction of
\[f \text{ is integrable} \iff \forall\varepsilon >0 \text{ } \exists\phi _ -, \phi _ + \text{ s.t. } I(\phi _ +) - I(\phi _ -) < \varepsilon\]
?
By the suprema and infimum approximation property respectively
\[\forall\varepsilon >0 \text{ } \exists \phi _ - \text{ s.t. } \sup I(\phi _ -) - \frac \varepsilon 2 < I(\phi_-)\]and
\[\forall\varepsilon >0 \text{ } \exists \phi _ + \text{ s.t. } I(\phi _ +) < \inf I(\phi_+) + \frac \varepsilon 2\]As these suprema and infima are assumed to be equal, we get that
\[\forall\varepsilon >0 \text{ } \exists\phi _ -, \phi _ + \text{ s.t. } I(\phi _ +) - I(\phi _ -) < \varepsilon\]@Prove the backward direction of
\[f \text{ is integrable} \iff \forall\varepsilon >0 \text{ } \exists\phi _ -, \phi _ + \text{ s.t. } I(\phi _ +) - I(\phi _ -) < \varepsilon\]
?
Let $\varepsilon > 0$ be arbitrary. Then
\[\begin{aligned} I(\phi _ +) - I(\phi _ -) < \varepsilon &\iff \inf I(\phi _ +) \le I(\phi _ +) < I(\phi _ -) + \varepsilon \le \sup I(\phi _ -) + \varepsilon \\ &\iff \inf I(\phi _ +) - \sup I(\phi _ -) < \varepsilon \\ &\iff \inf I(\phi _ +) = \sup I(\phi _ -) \end{aligned}\]Proofs
$f$ is integrable if
\[\sup _ {\phi _ -} I(\phi _ -) = \inf _ {\phi _ +} I(\phi _ +)\]
@Prove that this is true if and only if
\[\forall \varepsilon > 0 \text{ } \exists \phi _ -, \phi _ + \text{ s.t. } I(\phi _ +) - I(\phi _ -) < \varepsilon\]
Todo.