7Abstract Testing Protocols
In this chapter, we study supermartingales and global upper expected values in abstract testing protocols. We begin with the following protocol.
Protocol 7.1 specializes Protocol 6.12, in which 
is the only parameter and Forecaster can choose a different upper expectation on each round. The specialization fixes a single upper expectation 
and eliminates Forecaster by constraining him to use 
on every round. We have changed Reality's name to World. At first glance, Protocol 7.1 seems too special to serve as the setting for a widely applicable theory. But most of the results we obtain for it will extend directly to other testing protocols.
We study Protocol 7.1 in Sections 7.1–7.4. Section 7.1 reviews terminology and notation. Section 7.2 proves Doob's convergence theorem. Section 7.3 defines the upper expected value 
of a global variable 
in a situation and studies ...
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