 For each formula_17, let

and then, let

For each Borel measurable function formula_22 on formula_23, define, for each formula_24,

Since formula_26 and the mapping given by formula_27 is right continuous, we see that 
for any uniformly continuous function formula_28, we have the mapping given by formula_29 is right continuous.Therefore, together with the monotone class theorem, for any universally measurable function formula_28, the mapping given by formula_31, is jointly measurable, that is, formula_32 measurable, and subsequently, the mapping is also formula_33-measurable for all finite measures formula_34 on formula_35 and formula_36 on formula_16.Here, 
formula_33 is the completion of
formula_39 with respect
to the product measure formula_40.