\(A\)-integrable martingale sequences and application to Haar series (Q2774494)

A random variable (r.v.) \(X\) defined on a probability space \((\Omega, \mathcal B, P)\) is said to be \(A\)-integrable over a set \(B\in \mathcal B\) if NEWLINE\[NEWLINE P(\{\omega \in B: |X(\omega)|>C\}) = \overline {o}(1/C) \quad \text{as } C\to \infty, \tag{1} NEWLINE\]NEWLINE and if there exists a finite limit NEWLINE\[NEWLINE \lim_{C\to \infty} \int_{\{\omega\in B: |X(\omega)|\leq C\}} X dP = I. NEWLINE\]NEWLINE Then \(I\) is called the \(A\)-integral of \(X\) over \(B\) and is denoted by \((A)\int _B X dP\). NEWLINENEWLINENEWLINEA family of r.v. \(\{X_\gamma \}_{\gamma \in \Gamma }\), defined on \((\Omega, \mathcal B, P)\) (\(\Gamma \) is some index set) is said to be uniformly \(A\)-integrable on \(B\in \mathcal B\) iff the sets \(D_{\gamma }(C)=\{\omega \in B:|X_\gamma (\omega)|>C\}\) satisfy the conditions: \(P(D_{\gamma }(C))=\overline {o}(1/C)\) uniformly in \(\gamma \) as \(C\to \infty \) and NEWLINE\[NEWLINE \sup _{\gamma \in \Gamma }\Biggl |(A) \int _{D_{\gamma }(C)} X_\gamma dP\Biggr|\rightarrow 0\quad\text{as } C\to \infty. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIt is proved that the uniform \(A\)-integrability of partial sums of a Haar series convergent almost everywhere to an \(A\)-integrable function \(f\) is a sufficient condition for this series to be \(A\)-Fourier series of \(f\). At the same time there exists a Haar series which is everywhere convergent to an \(A\)-integrable function and which is not \(A\)-Fourier series of this function.