## Paper: The Scott Topology Induces the Weak Topology (at LICS 1996)

**Abbas Edalat**

### Abstract

Given a probability measure on a compact metric space, we construct an increasing chain of valuations on the upper space of the metric space whose least upper bound is the measure. We then obtain the expected value of any Hölder continuous function with respect to the measure up to any precision. We prove that the Scott topology induces the weak topology of the space of probability measures in the following general setting: Whenever a separable metric space is embedded into a subset of the maximal elements of an $\omega$-continuous dcpo, which is a $G_{\delta}$ subset of the dcpo equipped with the Scott topology, we show that the space of probability measures of the metric space equipped with the weak topology is then embedded into a subspace of the maximal elements of the probabilistic power domain of the dcpo. We present a novel application in the theory of periodic doubling route to chaos.

### BibTeX

@InProceedings{Edalat-TheScottTopologyInd, author = {Abbas Edalat}, title = {The Scott Topology Induces the Weak Topology}, booktitle = {Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS 1996)}, year = {1996}, month = {July}, pages = {372--381}, location = {New Brunswick, NJ, USA}, publisher = {IEEE Computer Society Press} }