In mathematics, specifically in measure theory, a **Borel measure** on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.

Let *X* be a locally compact Hausdorff space, and let
B
(
X
)
be the smallest σ-algebra that contains the open sets of *X*; this is known as the σ-algebra of Borel sets. A **Borel measure** is any measure *μ* defined on the σ-algebra of Borel sets. Some authors require in addition that *μ*(*C*) < ∞ for every compact set *C*. If a Borel measure *μ* is both inner regular and outer regular, it is called a **regular Borel measure** (some authors also require it to be tight). If *μ* is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure means *μ*(*C*) < ∞ for every compact set *C*.

The real line
R
with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case,
B
(
R
)
is the smallest σ-algebra that contains the open intervals of
R
. While there are many Borel measures μ, the choice of Borel measure which assigns
μ
(
(
a
,
b
]
)
=
b
−
a
for every half-open interval
(
a
,
b
]
is sometimes called "the" Borel measure on
R
. This measure turns out to be the restriction on the Borel σ-algebra of the Lebesgue measure
λ
, which is a complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the *completion* of the Borel σ-algebra, which means that it is the smallest σ-algebra which contains all the Borel sets and has a complete measure on it. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e.,
λ
(
E
)
=
μ
(
E
)
for every Borel measurable set, where
μ
is the Borel measure described above).

If *X* and *Y* are second-countable, Hausdorff topological spaces, then the set of Borel subsets
B
(
X
×
Y
)
of their product coincides with the product of the sets
B
(
X
)
×
B
(
Y
)
of Borel subsets of *X* and *Y*. That is, the Borel functor

B
o
r
:
T
o
p
2
C
H
a
u
s
→
M
e
a
s
from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.

The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.

One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral

(
L
μ
)
(
s
)
=
∫
[
0
,
∞
)
e
−
s
t
d
μ
(
t
)
.
An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function *f*. In that case, to avoid potential confusion, one often writes

(
L
f
)
(
s
)
=
∫
0
−
∞
e
−
s
t
f
(
t
)
d
t
where the lower limit of 0^{−} is shorthand notation for

lim
ε
↓
0
∫
−
ε
∞
.
This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

Given a Borel measure μ on a metric space *X* such that μ(*X*) > 0 and μ(*B*(*x*, *r*)) ≤ *r*^{s} holds for some constant *s* > 0 and for every ball *B*(*x*, *r*) in *X*, then the Hausdorff dimension dim_{Haus}(*X*) ≥ *s*. A partial converse is provided by Frostman's lemma:

**Lemma:** Let *A* be a Borel subset of **R**^{n}, and let *s* > 0. Then the following are equivalent:

*H*^{s}(*A*) > 0, where *H*^{s} denotes the *s*-dimensional Hausdorff measure.
There is an (unsigned) Borel measure *μ* satisfying *μ*(*A*) > 0, and such that
holds for all

*x* ∈

**R**^{n} and

*r* > 0.

The Cramér–Wold theorem in measure theory states that a Borel probability measure on
R
k
is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.