I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for the quality of its exercises. Unfortunately, some of them are marred with plain mistakes or wrong hints, and make me waste a lot of time detecting all of them. Does anybody know a reliable errata for the volume 2 of this treatise ?
Here below is one instance of a plain wrong hint in an exercise, and a sketch of a solution I found.
I would very much appreciate if somebody could give some kind of reassurance that I am on the right track here and that my solutions are correct.
Here is the text of the exercise:
The hint is obviously wrong because it is not possible to have the inequality $b\mu(A\cap F_q)\leq a\mu(A)$ for $q=n$ for instance.
Below is a solution I found for comments
Notation for $p\geq n\geq 0$ :
$$\begin{align*} A_{n}^p &=\left\{x\in X\ ;\ \sup_{p\geq r\geq n}f_{r}(x)\geq b\right\} \\ B_{n}^p &=\left\{x\in X\ ;\ \inf_{p\geq r\geq n}f_{r}(x)\leq a\right\} \\ \end{align*}$$
and
$$\begin{alignat*}{2} A_{n} &=\bigcup_{p\geq n} A_{n}^p &\quad B_{n} &=\bigcup_{p\geq n} B_{n}^p \\ A &=\bigcap_{n\geq 0}A_{n} &\quad B &=\bigcap_{n\geq 0}B_{n} \end{alignat*}$$
Then we have $E_{ab}=A\cap B$. We also note that the unions and intersections in the previous definitions are respectively increasing and decreasing sequences.
Choose $r\geq s\geq p\geq q\geq m\geq n$. First we notice that $$A_{n}^m = \bigcup_{m\geq i\geq n} \left\{x\in X\ ;\ f_{n}(x)<b, \cdots, f_{i-1}(x)<b,\ f_{i}(x)\geq b\right\}$$ the union being of disjoints sets. By definition of a martingale, for $i\leq m$, we have $$\int_{x\in X\ ;\ f_n(x)<\cdots,f_{i−1}(x)<b,\ f_i(x\geq b} f_i\ d\mu=\int_{x\in X\ ;\ f_n(x)<\cdots,f_{i−1}(x)<b,\ f_i(x\geq b}f_m\ d\mu$$ Then, we get $$\int_{A_n^m}f_{m}d\mu \geq b\mu(A_{n}^m)$$ For the same reasons, we also get $$\int_{B_{n}^m}f_{m}d\mu \leq a\mu(B_{n}^m)$$
Therefore, we deduce that $$\int_{A_{n}^m\cap B_{q}^p \cap A_{s}^r}f_{r} d\mu \geq b\mu(A_{n}^m\cap B_{q}^p \cap A_{s}^r)$$ and $$\int_{A_{n}^m\cap B_{q}^p \cap A_{s}^r}f_{r}d\mu \leq \int_{A_{n}^m\cap B_{q}^p}f_{r}d\mu =\int_{A_{n}^m\cap B_{q}^p}f_{p}d\mu \leq a\mu(A_{n}^m\cap B_{q}^p) \leq a\mu(A_{n}^m\cap B_{q})$$
Let $$b\mu(A_{n}^m\cap B_{q}^p \cap A_{s}^r) \leq a\mu(A_{n}^m\cap B_{q})$$
By successively having $r$ then $s$ then $p$ then $q$ goes towards infinity we get $$b\mu(A_{n}^m\cap B \cap A)\leq a\mu(A_{n}^m\cap B)$$ If $m$ then $n$ goes towards infinity we get $$b\mu(E_{ab})\leq a\mu(E_{ab})$$ QEA.
