I am reading a proof concerning a conditional density f ( t | G n ) where G n = σ ( T 1 , T 2 , . . . , T n ) is a sub σ-algebra generated by random variables T 1 , . . . , T n . In the proof, they express this conditional density in terms of the probability of T n + 1 being in some infinitesimal interval ds around t. They write: f ( t | G n ) = P ( T n + 1 ∈ [ t , t + d s ] | G n ) d s Intuitively, this makes sense to me, but I'm not really sure how to understand this more rigorously. I know that a conditioal probability given a σ-algebra is the same as a conditional expectation of an indicator function, but how do we make sense of this equality and the left hand side?Edit: f ( t | G n ) is the conditional density of event times in a point process given the first n points.

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I am reading a proof concerning a conditional density   f  (  t      |              G      n        ) where             G      n        =  σ  (      T    1    ,      T    2    ,  .  .  .  ,      T    n    ) is a sub   σ-algebra generated by random variables       T    1    ,  .  .  .  ,      T    n  . In the proof, they express this conditional density in terms of the probability of       T          n      +      1       being in some infinitesimal interval ds around t. They write:  f  (  t      |              G      n        )  =                    P            (              T                  n          +          1                    ∈      [      t      ,      t      +      d      s      ]              |                              G                n            )              d      s      Intuitively, this makes sense to me, but I&#039;m not really sure how to understand this more rigorously. I know that a conditioal probability given a   σ-algebra is the same as a conditional expectation of an indicator function, but how do we make sense of this equality and the left hand side?Edit:   f  (  t      |              G      n        ) is the conditional density of event times in a point process given the first n points.

Sydnee Villegas · Accepted Answer

You can define   f  (  t  ∣            G        n    ) as      lim          k      →      ∞                          P                    (        t        &amp;lt;                  T                      n            +            1                          ⩽        t        +                  δ          k                ∣                              G                    n                )                    δ      k        ,where   (      δ    k    ) is a deterministic sequence converging to 0, provided that the limit exists and is independent of the choice of the sequence   (      δ    k    ).

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