Frechet integral

Frechet integral - integral defined on the set of elements $F$ ${\ displaystyle F}$ $F$ arbitrary nature.

To determine the Frechet integral on the set $F$ ${\ displaystyle F}$ $F$ considered some $\sigma$ ${\ displaystyle \ sigma}$ $\ sigma$ - ring sets $T$ ${\ displaystyle T}$ $T$ with the set-additive function of a set given on it $\Phi (E)$ ${\ displaystyle \ Phi (E)}$ $\ Phi (E)$ with variations ${\overline {W}}(\Phi ,E)$ ${\ displaystyle {\ overline {W}} (\ Phi, E)}$ $\ overline {W} (\ Phi, E)$ and ${\underline {W}}(\Phi ,E)$ ${\ displaystyle {\ underline {W}} (\ Phi, E)}$ $\ underline {W} (\ Phi, E)$ . Let be $f(x)$ ${\ displaystyle f (x)}$ $f (x)$ - non-negative real function of the element $x$ ${\ displaystyle x}$ $x$ spaces $F$ ${\ displaystyle F}$ $F$ . Function $f(x)$ ${\ displaystyle f (x)}$ $f (x)$ called summable with respect to $\Phi$ ${\ displaystyle \ Phi}$ $\ Phi$ on set $E\subset T$ ${\ displaystyle E \ subset T}$ $E \ subset T$ if a series converges $\sum _{i}M_{i}W(\Phi ,E_{i})$ ${\ displaystyle \ sum _ {i} M_ {i} W (\ Phi, E_ {i})}$ $\ sum _ {{i}} M _ {{i}} W (\ Phi, E _ {{i}})$ with some partition of the set $E$ ${\ displaystyle E}$ $E$ on non-intersecting terms $E_{i}$ ${\ displaystyle E_ {i}}$ $E_ {i}$ , $E_{i}\subset T$ ${\ displaystyle E_ {i} \ subset T}$ $E _ {{i}} \ subset T$ , $M_{i}=\sup _{E_{i}}f$ ${\ displaystyle M_ {i} = \ sup _ {E_ {i}} f}$ $M _ {{i}} = \ sup _ {{E _ {{i}}}} f$ .

Frechet integral of the function $f(x)$ ${\ displaystyle f (x)}$ $f (x)$ defined as the difference of the integrals with respect ${\overline {W}}(\Phi ,E)$ ${\ displaystyle {\ overline {W}} (\ Phi, E)}$ $\ overline {W} (\ Phi, E)$ and ${\underline {W}}(\Phi ,E)$ ${\ displaystyle {\ underline {W}} (\ Phi, E)}$ $\ underline {W} (\ Phi, E)$ .

Necessary and sufficient conditions for the existence of a Frechet integral

In order to sum the function $f(x)$ ${\ displaystyle f (x)}$ was integrable in the sense of Frechet, it is necessary and sufficient that with any valid $a$ ${\ displaystyle a}$ lots of $E_{x}(f(x)>a)$ ${\ displaystyle E_ {x} (f (x)> a)}$ different from the set of $\sigma$ ${\ displaystyle \ sigma}$ -rings $T$ ${\ displaystyle T}$ on some subset of the set of measure zero belonging $\sigma$ ${\ displaystyle \ sigma}$ the ring.

Literature

Pesin I. N. The development of the concept of an integral. - M .: Science , 1980 . - 202 s.

Frechet integral

Necessary and sufficient conditions for the existence of a Frechet integral

Literature

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