## 10 February 2014

### Stopping time property of hitting times

Suppose that Ft, t ≥ 0, is an increasing family of σ-fields of subsets of a set Ω,  such that ⋂ε > 0 Ft = Ft, for all t ≥ 0.

Let F be a σ-field on Ω such that FtF for all t ≥ 0.

Let P be a probability measure on (Ω, F). Assume that each Ft contains every subset N of Ω  included in some set ANF with P(AN)=0.

Let, for each t ≥ 0, Xt be a measurable function from (Ω, Ft)  into a Polish space S, where the Polish space is equipped with the Borel σ-field B, i.e., the smallest σ-field containing all its open sets.

Assume that, for all ω ∈ Ω, the function tXt(ω) is continuous or right-continuous with discontinuities of first kind only.

Let BB, and define TB := inf{t ≥ 0: Xt ∈ B}. Then TB is a measurable function from Ω into [0, ∞] such that {ω ∈ Ω: TB(ω) ≤ t} ∈ Ft, for all t.

Probabilities and Potential, by Claude Dellacherie and Paul-André Meyer.

T H E B O T T O M L I N E

## What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

## The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

## The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant

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