1
$\begingroup$

"Note on the Riemann integral" in chapter 5 of Probability with Martingales by Williams reads:

If, for example, f is a non-negative Riemann integrable function on $([0,1],\mathcal{B}[0,1],Leb)$ with Riemann integral I, then there exists an increasing sequence of elements $(L_n)$ of elements of $SF^+$ and a decreasing sequence $(U_n)$ of elements of $SF^+$ such that

$L_n \uparrow L \leq f, U_n \downarrow U \geq f$


and $\mu(L_n) \uparrow I$,$\mu(U_n) \downarrow I$. If we define

$ \tilde{f} = \begin{cases} L & \text{ if $L=U$} \\ 0 & \text{otherwise} \\ \end{cases} $

then it is clear that $\tilde{f}$ is Borel measurable, while (since $\mu(L) = \mu(U) = 1$) $\{f \neq \tilde{f}\}$ is a subset of the Borel set $\{L \neq U\}$ which Lemma 5.2b showed to be of measure 0

Here, $SF^+$ is the collection of non-negative simple functions

and $\mu(f) :=: \int_{[0,1]} f(s)\mu(ds)$.

My question is: What are L and U?

  • I know $L(P,f)$ as the lower Riemann sum of $f$ using partition $P$
  • This would lead me to think that L would be the lower Riemann integral (the supremum of $L(P,f)$ taken over the partitions $P$)

BUT

  • The $L_n$ are functions, not numbers, and f is a function as well, so the convergence $L_n \uparrow L$ and inequality $L \leq f$ don't make sense.

So, what are $U$ and $L$?

Thank you very much in advance.

$\endgroup$
2
$\begingroup$

I believe what's meant is that the sequence of functions $L_n$ converges to a function $L$ with the property that $L(x) \le f(x)$ for every $x$, and similarly for the $U_n$.

That's certainly true, and seems to make sense in the context given.

As @Nate notes below, this convergence is probably meant to be understood pointwise.

$\endgroup$
  • $\begingroup$ That makes sense! Any idea on why $\mu(L) = 1$? Or should that be $\mu(L) = I$? $\endgroup$ – user2449397 Aug 26 '14 at 15:21
  • $\begingroup$ Almost certainly $\mu(L) = I$, since they've earlier established that $\mu(L_n) \to I$. I think that the "1" is just a typo. $\endgroup$ – John Hughes Aug 26 '14 at 15:24
1
$\begingroup$

Unless otherwise specified, statements about convergence, inequalities, etc, of functions are usually meant to be interpreted pointwise. So "$L_n \uparrow L$" means $L_n(x) \uparrow L(x)$ for every $x$, and "$L \le f$" means $L(x) \le f(x)$ for every $x$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.