Bemerkungen zu den Sätzen von Jordan (Q792687)

Let (\(\Omega,{\mathcal A},\mu)\) be a measure space and \(A_ i\in {\mathcal A}\) with the indicators \(I_ i\) for \(i=1,2,...\), further \(X:=\sum_{i}I_ i\). We ask for \(\mu(X\geq m)\) and \(\mu(X=m)\), \(m=1,2,...\). Clearly, if \(\mu\) is a probability measure then \(\mu(X\geq m)\) \((\mu(X=m))\) is the probability that at least (exactly) m among the events \(A_ 1,A_ 2,..\). occur, and in case \(A_ i=\emptyset\) for \(i=n+1,n+2,...,m<n\) the well-known Jordan-theorems hold. In the present paper generalized Jordan- theorems are derived. Necessary and sufficient conditions are given for the representation of \(\mu(X\geq m)\), \(\mu(X=m)\) by infinite series in the terms \( M_ j:=\sum_{N\subset {\mathbb{N}},| N| =j}\mu(\cap_{i\in N}A_ i),\quad j=m,m+1,....\) Estimations of the residual terms are given, too.