Материал из Викиверситета
Пусть
μ
{\displaystyle \mu }
ν
{\displaystyle \nu }
f
{\displaystyle f}
μ
⊗
ν
{\displaystyle \mu \otimes \nu }
μ
{\displaystyle \mu }
x
{\displaystyle x}
y
↦
f
(
x
,
y
)
{\displaystyle y\mapsto f(x,y)}
ν
{\displaystyle \nu }
ν
{\displaystyle \nu }
y
{\displaystyle y}
x
↦
f
(
x
,
y
)
{\displaystyle x\mapsto f(x,y)}
μ
{\displaystyle \mu }
x
↦
∫
Y
f
(
x
,
y
)
ν
(
d
y
)
{\displaystyle x\mapsto \int _{Y}f(x,y)\nu (dy)}
y
↦
∫
X
f
(
x
,
y
)
μ
(
d
x
)
{\displaystyle y\mapsto \int _{X}f(x,y)\mu (dx)}
∫
X
×
Y
f
d
(
μ
⊗
ν
)
=
∫
Y
∫
X
f
(
x
,
y
)
μ
(
d
x
)
ν
(
d
y
)
=
=
∫
X
∫
Y
f
(
x
,
y
)
ν
(
d
y
)
μ
(
d
x
)
{\displaystyle {\begin{aligned}\int _{X\times Y}fd(\mu \otimes \nu )&=\int _{Y}\int _{X}f(x,y)\mu (dx)\nu (dy)=\\&=\int _{X}\int _{Y}f(x,y)\nu (dy)\mu (dx)\end{aligned}}}
