在测度论中，可测函数 （measurable function）是研究的重要对象，就像数学分析主要研究连续函数 那样。
实变函数（Lebesgue 测度 ）上的可测函数参见可测函数 。
定义 [ ] 假设有可测空间
(
X
,
M
)
{\displaystyle (X, \mathcal{M})}
(
R
¯
,
B
R
¯
)
{\displaystyle (\overline{\R}, \mathcal{B}_{\overline{\R}})}
R
¯
{\displaystyle \overline{\R}}
广义实数
±
∞
{\displaystyle \pm\infty}
Borel 域 形成的可测空间，映射
f
(
x
)
:
X
→
R
¯
{\displaystyle f(x): X \to \overline{\R}}
f
{\displaystyle f}
{
x
∈
X
:
f
(
x
)
>
t
}
∈
M
,
∀
t
∈
R
.
{\displaystyle \{ x \in X: f(x) > t \} \in \mathcal{M}, \forall t \in \R.}
{
x
∈
X
:
f
(
x
)
⩽
t
}
∈
M
,
∀
t
∈
R
.
{\displaystyle \{ x \in X: f(x) \leqslant t \} \in \mathcal{M}, \forall t \in \R.}
{
x
∈
X
:
f
(
x
)
⩾
t
}
∈
M
,
∀
t
∈
R
.
{\displaystyle \{ x \in X: f(x) \geqslant t \} \in \mathcal{M}, \forall t \in \R.}
{
x
∈
X
:
f
(
x
)
<
t
}
∈
M
,
∀
t
∈
R
.
{\displaystyle \{ x \in X: f(x) < t \} \in \mathcal{M}, \forall t \in \R.}
在概率论中，可测函数也被称为随机变量 。
性质 [ ] 运算性质 [ ] 可测函数有如下运算性质：
M
{\displaystyle \mathcal{M}}
f
(
x
)
{\displaystyle f(x)}
c
f
(
x
)
{\displaystyle cf(x)}
M
{\displaystyle \mathcal{M}}
f
(
x
)
,
g
(
x
)
{\displaystyle f(x), g(x)}
f
(
x
)
±
g
(
x
)
{\displaystyle f(x) \pm g(x)}
M
{\displaystyle \mathcal{M}}
f
(
x
)
,
g
(
x
)
{\displaystyle f(x), g(x)}
f
(
x
)
g
(
x
)
{\displaystyle f(x) g(x)}
M
{\displaystyle \mathcal{M}}
f
(
x
)
,
g
(
x
)
{\displaystyle f(x), g(x)}
f
(
x
)
/
g
(
x
)
{\displaystyle f(x)/g(x)}
可测函数的极限运算性质：设
{
f
k
(
x
)
}
k
=
1
∞
{\displaystyle \{ f_k (x) \}_{k=1}^\infty}
M
{\displaystyle \mathcal{M}}
sup
k
⩾
1
f
k
(
x
)
,
inf
k
⩾
1
f
k
(
x
)
{\displaystyle \sup_{k \geqslant 1} f_k(x), \inf_{k \geqslant 1} f_k (x)}
M
{\displaystyle \mathcal{M}}
lim sup
k
→
∞
f
k
(
x
)
,
lim inf
k
→
∞
f
k
(
x
)
{\displaystyle \limsup_{k \to \infty} f_k(x), \liminf_{k \to \infty} f_k (x)}
M
{\displaystyle \mathcal{M}}
简单函数 [ ] 假设有可测空间
(
X
,
M
)
{\displaystyle (X, \mathcal{M})}
{
E
k
}
k
=
1
n
{\displaystyle \{ E_k \}_{k=1}^n}
X
=
⋃
k
=
1
n
E
k
{\displaystyle X = \bigcup_{k=1}^n E_k}
{
E
k
}
{\displaystyle \{ E_k \}}
X
{\displaystyle X}
{
E
k
}
⊂
M
{\displaystyle \{ E_k \} \subset \mathcal{M}}
{
E
k
}
{\displaystyle \{ E_k \}}
如果存在
X
{\displaystyle X}
{
E
k
}
{\displaystyle \{ E_k \}}
{
a
k
}
{\displaystyle \{ a_k \}}
f
:
X
→
R
{\displaystyle f: X \to \R}
f
(
x
)
=
∑
k
=
1
n
a
k
I
E
k
(
x
)
{\displaystyle f(x) = \sum_{k=1}^n a_k I_{E_k}(x)}
f
{\displaystyle f}
f
{\displaystyle f}
对于简单函数，它可以逼近可测函数，即有如下的简单函数逼近定理：
设
f
{\displaystyle f}
X
{\displaystyle X}
函数列
{
f
k
(
x
)
}
k
=
1
∞
:
f
1
(
x
)
⩽
f
2
(
x
)
⩽
⋯
⩽
f
k
(
x
)
⩽
⋯
{\displaystyle \{ f_k (x) \}_{k=1}^\infty: f_1(x) \leqslant f_2(x) \leqslant \cdots \leqslant f_k(x) \leqslant \cdots}
lim
k
→
∞
f
k
(
x
)
=
f
(
x
)
,
x
∈
X
.
{\displaystyle \lim_{k \to \infty} f_k (x) = f(x), \quad x \in X.}
设
f
{\displaystyle f}
X
{\displaystyle X}
{
f
k
(
x
)
}
k
=
1
∞
:
|
f
k
(
x
)
|
⩽
f
(
x
)
{\displaystyle \{ f_k (x) \}_{k=1}^\infty: |f_k(x)| \leqslant f(x)}
lim
k
→
∞
f
k
(
x
)
=
f
(
x
)
,
x
∈
X
.
{\displaystyle \lim_{k \to \infty} f_k (x) = f(x), \quad x \in X.}
f
k
(
x
)
{\displaystyle f_k(x)}
一致收敛 。 参考资料 Lawrence C. Evans, Ronald F. Gariepy, Measure Theory and Fine Properties of Functions(4th Ed.) , Studies in Advanced Mathematics Vol.5 , CRC Press, 1991, ISBN 978-0-8493-7157-8 
