The present paper is devoted to the study of the direct limits of direct systems of measure (resp. probability) spaces. If I is a right directed preordered set, (E[alpha])[alpha][set membership, variant]I a family of sets indexed by I, G = [up curve][alpha][set membership, variant]I E[alpha] - {[alpha]} the sum of the family (E[alpha]), [alpha] a [sigma]-algebra in E[alpha] for each [alpha][set membership, variant]I and M = [up curve][alpha][set membership, variant]I[alpha] - {{[alpha]}} is the sum of the family ([alpha]), then it is shown that M is a [sigma]-algebra in G. If is the direct limit of the family (E[alpha]), if (E[alpha]) the direct limit of the family of power sets ((E[alpha])), if = lim [alpha] is the direct limit of the family ([alpha]), if (E[alpha], [alpha]) is a direct system of measurable spaces, then () is a measurable space. If ([lambda][alpha])[alpha][set membership, variant]I is a direct system of measures with values in a complete abelian group, if is the direct limit of the family ([lambda][alpha]), and if (E[alpha], [alpha], [lambda][alpha]) is a direct system of measure (resp. probability) spaces, then it is shown that the direct limit () is a measure (resp. probability) space. Further papers will be devoted to the applications of these direct limits in the measure (resp. probability) theory.
