I am not that familiar in proving measurability of a function. The definition I know is that the preimage of measurable sets has to be measurable again. But how to prove that seems difficult to me at the first glance. I do have the following exercise: Let ( B t ) be a Brownian motion on ( Ω , A , P ) with continuous trajectories/paths. Consider the product space ( [ 0 , ∞ ) , B [ 0 , ∞ ) , λ ) ⊗ ( Ω , A , P )where λ is the Lebesgue measure. Show that the mapping ( t , ω ) → ( B t ( ω ) )( t ≥ 0) defined from that space to ( R [ 0 , ∞ ) , B R [ 0 , ∞ ) ) is measurable. How can I prove this? I think the continuity is important, but I dont know how to involve it. Any help is appreciated.

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I am not that familiar in proving measurability of a function. The definition I know is that the preimage of measurable sets has to be measurable again. But how to prove that seems difficult to me at the first glance. I do have the following exercise: Let   (      B    t    ) be a Brownian motion on   (  Ω  ,      A    ,      P    ) with continuous trajectories/paths. Consider the product space  (  [  0  ,  ∞  )  ,            B              [      0      ,      ∞      )        ,  λ  )  ⊗  (  Ω  ,      A    ,      P    )where   λ is the Lebesgue measure. Show that the mapping   (  t  ,  ω  )  →  (      B    t    (  ω  )  )(  t  ≥  0) defined from that space to   (            R              [      0      ,      ∞      )        ,            B                                R                          [          0          ,          ∞          )                      ) is measurable. How can I prove this? I think the continuity is important, but I dont know how to involve it. Any help is appreciated.

Arcatuert3u · Accepted Answer

Approximate   B  (  t  ,  ω  ) by the sequence of simple processes   {      B          (      n      )        (  t  ,  ω  )  =  B  (      2          −      n        ⌊      2          n        t  ⌋  ,  ω  )  }. It is easy to see that   (  t  ,  ω  )  ↦      B          (      n      )        (  t  ,  ω  ) is             B              [      0      ,      ∞      )        ⊗      A  -measurable and       B          (      n      )        →  B pointwise as   n  →  ∞ (  ∵ B is continuous). Thus,   B is also measurable as the limit of measurable functions.

I am not that familiar in proving measurability of a function. The definition I know is that the pre

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