# A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said to be "apparent". The problem goes as the following :

Let $X$ and $Y$ be locally compact Hausdorff spaces. Let $M(X)$ denote the space of all regular complex (hence bounded) measures on $X$, and $M^+(X)$ denote the space of all non-negative measures in $M(X)$.

We equip $M^+(X)$ with the cone topology, i.e, the weak topology induced by the family of functionals $\{\mu \mapsto \int_X f d\mu:f \in C_c^+(X)\cup \{1_X\} \}$, where $C^+_c(X)$ is the set of all non-negative compactly supported continuous functions on $X$ and $1_X$ is the characteristic function on $X$. Similarly we have spaces of measures $M(Y), M^+(Y)$.

We have a linear function $\phi : M(X) \rightarrow M(Y)$ given by $\mu \mapsto \mu'$ such that

(i) If $\mu \geq 0$ then $\mu' \geq 0$.

(ii) The restricted mapping $\phi|_{M^+(X)}$ from $M^+(X)$ to $M^+(Y)$ is continuous in the cone topology.

Let $p_x$ be the point-mass measure at $x \in X$. Also, suppose that $m$ is a non-negative measure on $X$ and $g$ is a lower semi-continuous function on $Y$.

Define a measure $m'$ on $Y$ by $m' := \int_X p_x' \ dm(x)$, and a function $g'$ on $X$ by $g'(x) := \int_Y g(y) \ dp_x'(y)$.

Now it is apparent from the definitions that $\int_Xg' \ dm = \int_Y g \ dm'$.

I need to show that $g'$ is lower semi-continuous on $X$.

What I already have is :

(a) The number $M := \sup_{x \in X} ||p_x'||$ is finite.

(b) If $h$ is a bounded continuous function on $Y$, then $h'$ is continuous and $||h'||_\infty \leq M ||h||_\infty$.

(c) The function $x \mapsto p_x$ is a homeomorphism from $X$ onto a closed subset of $M^+(X)$.

Any kind of help / comments will be Really appreciated ! :)

• This was also posted to MathOverflow. Please note that crossposting between SE sites is highly frowned upon - try one site first, and if you don't get a satisfactory response, ask a moderator to migrate the question to a different site. If you insist on posting in many sites, at least provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to find out that you'd already gotten an answer elsewhere. – Zev Chonoles Oct 16 '13 at 9:44
• Okay, thanks for the info @ZevChonoles . Actually I am new here, and did not know that these two sites are connected. Should I withdraw the question from one of the sites ? – NewUser Oct 16 '13 at 9:48
• The way you've defined $m'$ doesn't quite make sense; do you mean $m'(A) = \int_A p'_x\,dm(x)$? – Nate Eldredge Oct 16 '13 at 15:16
• @NateEldredge No, it means that $m'(A) = \int_X p_x'(A) \ dm(x)$. It is given that this $m'$ exists. – NewUser Oct 16 '13 at 15:20
• Also, the topology on $M^+(X)$ is the cone topology, i.e, the weak topology induced by the family of functionals $\{ \mu \mapsto \int_X f \ d\mu : f \in C_c^+(X) \cup \{1_X\} \ \}$, where $C_c^+(X)$ is the set of all non-negative compactly supported continuous functions on $X$ and $1_X$ is the characteristic function on $X$. – NewUser Oct 16 '13 at 15:23