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Let $X$ be a Polish space, that is, a completely metrizable separable space. We say that $f:X \rightarrow \mathbb{R}$ is a Baire class $1$ function if for every real number $c,$ the two sets $\{ f \leq c \}$ and $\{ f \geq c\}$ are $F_{\sigma}$ (countable union of closed sets).

In this paper, the author define the following definition in page $8129$:

For a Baire class $1$ function, define $$\alpha'_1(f) = \min\{ \max\{ \alpha_1(f_1),..., \alpha_1(f_n) \} : n \text{ is finite },$$ $$f=\sum_{i=1}^nf_i \text{ for some Baire class 1 functions }f_1,f_2,...,f_n \}$$

Question: The authors claimed that $\alpha'_1$ is essentially linear on the set of Baire class $1$ functions, that is, for every pair of Baire class $1$ functions $f$ and $g$, we have $\alpha_1'(f+g) \leq \max\{ \alpha_1'(f), \alpha_1'(g) \}.$

I have no idea how to prove it. Any hint would be appreciated.

Definition: Let $D$ be a derivative. The rank of $D$ is the smallest ordinal $\eta$ such that $D^{\eta}(X) = \emptyset,$ if such ordinal exists; $\omega_1$ otherwise. Denote the rank of $D$ by rk$(D).$

Definition: Let $A$ and $B$ be two subsets of $X.$ We associate a derivative with them by $$D_{A,B}(F) = \overline{F \cap A} \cap \overline{F \cap B}.$$ Denote $\alpha(A,B) = \text{rk}(D_{A,B}).1$

Definition: The separation rank of a Baire class $1$ function $f$ s defined as $$\alpha(f) = \sup_{p<q , p,q \in \mathbb{Q}}\alpha(\{ f \leq p \}, \{ f \geq q \}).$$

Definition: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of the shortest such sequence; otherwise let $\alpha_1(A,B) = \omega_1.$ We define modified separation rank of a Baire class $1$ function $f$ as $$\alpha_1(f) = \sup_{p < q, p,q \in \mathbb{Q}}\alpha_1(\{ f \leq p\}, \{f \geq q\}).$$

Let $(F_{\eta})_{\eta < \lambda}$ be a (not necessarily distinct) decreasing sequence of sets. Let us assume that $F_0 = X$ and that the sequence is continuous, that is, $F_{\eta} = \cap_{\theta < \eta}F_{\theta}$ for every limit $\eta$ and if $\lambda$ is a limit, then $\cap_{\eta < \lambda}F_{\eta} = \emptyset.$ If $\eta \geq \lambda,$ then $F_{\eta} = \emptyset.$ We say that $H$ is the transfinite difference of $(F_{\eta})_{\eta < \lambda}$ if $H = \cup_{\eta < \lambda, \eta \text{ even}}(F_{\eta} \setminus F_{\eta +1}).$

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  • $\begingroup$ What is $\alpha_1$? $\endgroup$ – Andrés E. Caicedo Jul 15 '17 at 0:28
  • $\begingroup$ Included defintion of $\alpha_1$. $\endgroup$ – Idonknow Jul 15 '17 at 0:33
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:-) Let $$\alpha'_1(f) =\max\{ \alpha_1(f_1), \dots, \alpha_1(f_n) \}$$ and $$\alpha'_1(g) =\max\{ \alpha_1(g_1), \dots, \alpha_1(g_m) \}$$ for some Baire class 1 functions $f_1, f_2,\dots, f_n,g_1,\dots, g_m$ such that $$f=\sum_{i=1}^nf_i$$ and $$g=\sum_{j=1}^mg_j.$$ Then $$f+g=\sum_{i=1}^nf_i +\sum_{j=1}^mg_j,$$ so $$\alpha'_1(f+g)\le \max\{ \alpha_1(f_1),\dots, \alpha_1(f_n), \alpha_1(g_1),\dots, \alpha_1(g_m) \}=\max \{\max\{ \alpha_1(f_1),\dots, \alpha_1(f_n) \}, \max\{ \alpha_1(g_1),\dots, \alpha_1(g_m) \}\}=$$ $$\max\{ \alpha_1'(f), \alpha_1'(g) \}.$$

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