The name is absent

Alternatively, we can apply Bayesian updating to μ_t, yielding

Now we observe that, since E ∩ E_t+_i Ç E_t+_i Ç E_t,

and hence,

μ^μ(e∩^Et+i⁾ʌ , μ^μ‰η

∖   μ (Et)   Γ∖ μ (Et)J

= ^μt+i ^(E) ■

6 Restricted Bayesianism

Given full rationality on a bounded domain, it is natural to consider μ^l_t, the
restriction of the probability measure μ_t to events E (p) where p E N^г. That
^is,

μt^{(E (}p⁾⁾ = μt^{(E (}p⁾⁾

if and only if p E N^t.

We now have two potential ways of deriving μ^l_t+i, given the observation
of E_t+_i. We can use the restriction procedure at time t + 1 instead of t,
obtaining μt_+i as the restriction of μ_t+i to P^t. Alternatively, we can apply
Bayesian updating directly to μt using the information obtained from Et+_i.
The first approach yields, for any p E P^t,

μ^lt+i^{(E (}p⁾⁾ = μt+i^{(E (}p⁾⁾

= μ^{(E (}p⁾ ∩ ^e

^μt ^(Et+i⁾

= μ (E (p) ∩ Et+i)

^{μ (E}t+i⁾

where, as shown in the previous section, the last step works because E (p) ∩
^Et+i ^{Ç E}t+i ^{Ç E}t^so μ ^(Et+i⁾ = μ_t ^(Et+i⁾ μ ^(Et⁾

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