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 ! :)