Convex measure
Open-knowledge reference entry
In measure and probability theory in mathematics, a convex measure is a probability measure that — loosely put — does not assign more mass to any intermediate set "between" two measurable sets A and B than it does to A or B individually. There are multiple ways in which the comparison between the probabilities of A and B and the intermediate set can be made, leading to multiple definitions of convexity, such as log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex spaces in the 1970s.
General definition and special cases
Let X be a locally convex Hausdorff vector space, and consider a probability measure μ on the Borel σ-algebra of X. Fix −∞ ≤ s ≤ 0, and define, for u, v ≥ 0 and 0 ≤ λ ≤ 1,
M
s
,
λ
(
u
,
v
)
=
{
(
λ
u
s
+
(
1
−
λ
)
v
s
)
1
/
s
if
−
∞
<
s
<
0
,
min
(
u
,
v
)
if
s
=
−
∞
,
u
λ
v
1
−
λ
if
s
=
0.
{\displaystyle M_{s,\lambda }(u,v)={\begin{cases}(\lambda u^{s}+(1-\lambda )v^{s})^{1/s}&{\text{if }}-\infty <s<0,\\\min(u,v)&{\text{if }}s=-\infty ,\\u^{\lambda }v^{1-\lambda }&{\text{if }}s=0.\end{cases}}}
For subsets A and B of X, we write
λ
A
+
(
1
−
λ
)
B
=
{
λ
x
+
(
1
−
λ
)
y
∣
x
∈
A
,
y
∈
B
}
{\displaystyle \lambda A+(1-\lambda )B=\{\lambda x+(1-\lambda )y\mid x\in A,y\in B\}}
for their Minkowski sum. With this notation, the measure μ is said to be s-convex if, for all Borel-measurable subsets A and B of X and all 0 ≤ λ ≤ 1,
μ
(
λ
A
+
(
1
−
λ
)
B
)
≥
M
s
,
λ
(
μ
(
A
)
,
μ
(
B
)
)
.
{\displaystyle \mu (\lambda A+(1-\lambda )B)\geq M_{s,\lambda }(\mu (A),\mu (B)).}
The special case s = 0 is the inequality
μ
(
λ
A
+
(
1
−
λ
)
B
)
≥
μ
(
A
)
λ
μ
(
B
)
1
−
λ
,
{\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },}
i.e.
log
μ
(
λ
A
+
(
1
−
λ
)
B
)
≥
λ
log
μ
(
A
)
+
(
1
−
λ
)
log
μ
(
B
)
.
{\displaystyle \log \mu (\lambda A+(1-\lambda )B)\geq \lambda \log \mu (A)+(1-\lambda )\log \mu (B).}
Thus, a measure being 0-convex is the same thing as it being a logarithmically concave measure.
Properties
The classes of s-convex measures form a nested increasing family as s decreases to −∞"
s
≤
t
and
μ
is
t
-convex
⟹
μ
is
s
-convex
{\displaystyle s\leq t{\text{ and }}\mu {\text{ is }}t{\text{-convex}}\implies \mu {\text{ is }}s{\text{-convex}}}
or, equivalently
s
≤
t
⟹
{
s
-convex measures
}
⊇
{
t
-convex measures
}
.
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