在數學中，變差 （variation）是一個衡量函數某種可微性質的概念。關於函數和測度 的變差有著名的 Hahn-Jordan 分解 。
一維情形 [ ] 設
f
(
x
)
{\displaystyle f(x)}
區間
[
a
,
b
]
{\displaystyle [a, b]}
Δ
:
a
=
x
0
<
x
1
<
x
2
<
⋯
<
x
n
=
b
{\displaystyle \Delta: a = x_0 < x_1 < x_2 < \cdots < x_n = b}
v
Δ
=
∑
k
=
1
n
|
f
(
x
k
)
−
f
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x
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−
1
)
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{\displaystyle v_\Delta = \sum_{k=1}^n | f(x_k) - f(x_{k-1}) |}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a, b]}
變差 ，而
⋁
a
b
(
f
)
:=
sup
v
Δ
.
{\displaystyle \bigvee_a^b (f) := \sup v_\Delta.}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a, b]}
全變差 （total variation），即取所有變差的上確界。如果
⋁
a
b
(
f
)
<
+
∞
,
{\displaystyle \bigvee_a^b (f) < + \infty,}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a, b]}
有界變差函數 。
測度情形 [ ] 假設有可測空間
(
X
,
F
)
{\displaystyle (X, \mathcal{F})}
符號測度
φ
{\displaystyle \varphi}
測度
φ
+
{\displaystyle \varphi^+}
φ
−
{\displaystyle \varphi^-}
φ
=
φ
+
−
φ
−
.
{\displaystyle \varphi = \varphi^+ - \varphi^-.}
φ
+
=
φ
∗
,
φ
−
=
(
−
φ
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∗
.
{\displaystyle \varphi^+ = \varphi^*, \varphi^- = (-\varphi)^*.}
φ
∗
(
A
)
:=
sup
{
φ
(
B
)
:
B
⊂
A
,
B
∈
F
}
.
{\displaystyle \varphi^*(A) := \sup \{ \varphi(B): B \subset A, B \in \mathcal{F} \}.}
φ
{\displaystyle \varphi}
測度
φ
+
,
φ
−
{\displaystyle \varphi^+, \varphi^-}
φ
{\displaystyle \varphi}
|
φ
|
:=
φ
+
+
φ
−
{\displaystyle |\varphi| := \varphi^+ + \varphi^-}
複測度情形 [ ] 假設可測空間
(
X
,
F
)
{\displaystyle (X, \mathcal{F})}
ν
{\displaystyle \nu}
f
{\displaystyle f}
ν
{\displaystyle \nu}
ν
{\displaystyle \nu}
變差 ，選擇適當的
(
X
,
F
)
{\displaystyle (X, \mathcal{F})}
μ
{\displaystyle \mu}
f
∈
L
1
(
X
,
μ
)
{\displaystyle f \in L^1(X, \mu)}
ν
(
E
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=
∫
E
f
d
μ
,
∀
E
∈
F
.
{\displaystyle \nu(E) = \int_E f \mathrm{d}\mu, \quad \forall E \in \mathcal{F}.}
|
ν
|
(
E
)
:=
∫
E
|
f
|
d
μ
.
{\displaystyle |\nu|(E) := \int_E |f| \mathrm{d}\mu.}
ν
{\displaystyle \nu}
變差 。這裡需要說明的是：上述定義雖然用到了
μ
,
f
{\displaystyle \mu, f}
當
ν
{\displaystyle \nu}
Hahn-Jordan 分解 定義出全變差的概念等價。
等價定義 [ ]
|
ν
|
{\displaystyle |\nu|}
∀
E
∈
F
:
{\displaystyle \forall E \in \mathcal{F}:}
|
ν
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sup
{
∑
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1
n
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ν
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:
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n
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i
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∅
,
n
⩾
N
}
.
{\displaystyle |\nu|(E) = \sup \left\{ \sum_{i=1}^n |\nu(E_i)|: E = \bigcup_{i=1}^n E_i, \bigcap_{i=1}^n E_i = \varnothing, n \geqslant \N \right\}.}
|
ν
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(
E
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=
sup
{
∑
i
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1
∞
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ν
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i
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:
E
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⋃
i
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∞
E
i
,
⋂
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∞
E
i
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∅
}
.
{\displaystyle |\nu|(E) = \sup \left\{ \sum_{i=1}^\infty |\nu(E_i)|: E = \bigcup_{i=1}^\infty E_i, \bigcap_{i=1}^\infty E_i = \varnothing \right\}.}
|
ν
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(
E
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=
sup
{
|
∫
E
f
d
μ
|
:
‖
f
‖
L
1
(
X
,
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)
⩽
1
}
.
{\displaystyle |\nu|(E) = \sup \left\{ \left|\int_E f \mathrm{d}\mu \right|: \| f \|_{L^1(X, \mu)} \leqslant 1 \right\}.}
μ
{\displaystyle \mu}
(
X
,
F
)
{\displaystyle (X, \mathcal{F})}
f
∈
L
1
(
X
,
μ
)
.
{\displaystyle f \in L^1(X, \mu).}
性質 [ ] 假設可測空間
(
X
,
F
)
{\displaystyle (X, \mathcal{F})}
ν
,
μ
{\displaystyle \nu, \mu}
∀
E
∈
F
,
|
ν
(
E
)
|
⩽
|
μ
|
(
E
)
.
{\displaystyle \forall E \in \mathcal{F}, |\nu(E)| \leqslant |\mu|(E).}
ν
{\displaystyle \nu}
|
ν
|
{\displaystyle |\nu|}
d
ν
d
|
ν
|
{\displaystyle \dfrac{\mathrm{d} \nu}{\mathrm{d} |\nu|}}
|
ν
|
{\displaystyle |\nu|}
f
∈
L
1
(
ν
)
{\displaystyle f \in L^1(\nu)}
f
∈
L
1
(
|
ν
|
)
{\displaystyle f \in L^1(|\nu|)}
|
∫
E
f
d
ν
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⩽
∫
E
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f
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d
|
ν
|
,
∀
f
∈
L
1
(
ν
)
.
{\displaystyle \left| \int_E f \mathrm{d}\nu \right| \leqslant \int_E |f| \mathrm{d}|\nu|, \quad \forall f \in L^1(\nu).}
|
ν
+
μ
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⩽
|
ν
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+
|
μ
|
.
{\displaystyle |\nu + \mu| \leqslant |\nu| + |\mu|.}
ν
=
|
ν
|
{\displaystyle \nu = |\nu|}
ν
(
X
)
=
|
ν
|
(
X
)
.
{\displaystyle \nu(X) = |\nu|(X).}
向量值測度情形 [ ] 像數值測度有全變差那樣，向量值測度也有（全）變差。同時數值測度有上下變差，向量值測度也有對應的半變差。
全變差 [ ] 假設
F
:
R
→
X
{\displaystyle F: \mathcal{R} \to X}
E
∈
R
{\displaystyle E \in \mathcal{R}}
E
{\displaystyle E}
R
{\displaystyle \mathcal{R}}
π
{\displaystyle \pi }
|
F
|
(
E
)
:=
sup
π
∑
A
∈
π
‖
F
(
A
)
‖
.
{\displaystyle |F|(E) := \sup_\pi \sum_{A \in \pi} \| F(A) \|.}
F
{\displaystyle F}
E
{\displaystyle E}
|
F
|
(
Ω
)
<
+
∞
{\displaystyle |F|(\varOmega) < +\infty}
F
{\displaystyle F}
如果
F
{\displaystyle F}
|
F
|
{\displaystyle |F|}
R
{\displaystyle \mathcal{R}}
半變差 [ ] 假設
F
:
R
→
X
{\displaystyle F: \mathcal{R} \to X}
E
∈
R
{\displaystyle E \in \mathcal{R}}
‖
F
‖
(
E
)
:=
sup
{
|
f
∘
F
|
(
E
)
:
f
∈
X
∗
,
‖
f
‖
⩽
1
}
{\displaystyle \| F \|(E) := \sup\{ |f \circ F| (E): f \in X^*, \|f\| \leqslant 1 \}}
F
{\displaystyle F}
‖
F
‖
(
Ω
)
<
+
∞
{\displaystyle \|F\|(\varOmega) < +\infty}
F
{\displaystyle F}
我們也希望用全變差中分割的方法描述半變差，這就得到半變差的下述等價定義：記
E
{\displaystyle E}
R
{\displaystyle \mathcal{R}}
π
{\displaystyle \pi }
‖
F
‖
(
E
)
=
sup
π
{
‖
∑
A
n
∈
π
α
n
F
(
A
n
)
‖
:
‖
α
n
‖
⩽
1
}
.
{\displaystyle \|F\|(E) = \sup_\pi \left\{ \left\| \sum_{A_n \in \pi} \alpha_n F(A_n) \right\|: \| \alpha_n \| \leqslant 1 \right\}.}
對任意可測集列
{
E
n
}
n
=
1
∞
{\displaystyle \{ E_n \}_{n=1}^\infty}
‖
F
‖
(
⋃
n
=
1
∞
E
n
)
⩽
∑
n
=
1
∞
‖
F
‖
(
E
n
)
.
{\displaystyle \|F\| \left( \bigcup_{n=1}^\infty E_n \right) \leqslant \sum_{n=1}^\infty \|F\|(E_n).}
{
∑
n
=
1
N
‖
F
‖
(
E
n
)
}
{\displaystyle \left\{ \sum_{n=1}^N \|F\|(E_n) \right\}}
範數 收斂之極限 。 參考資料 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 程士宏, 《測度論與概率論基礎》, 北京大學出版社, 北京, 2006-06, ISBN 978-7-3010-6345-3 
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