When I first studied probability and statistics many decades ago, a probability course based on William Feller's "An Introduction to Probability Theory and Its Applications, Volume 1" was prerequisite to the statistics course that used Hogg and Craig. With Volume 1's restriction to finite (maybe also countable; I forget) sample spaces, Feller achieved considerable rigor without needing measure theory. Even in Volume 2, his use of measure theory was generally approachable. Both Volumes are still available, over 40 years since the author's death.

When I first studied probability and statistics many decades ago, a probability course based on William Feller's "An Introduction to Probability Theory and Its Applications, Volume 1" was prerequisite to the statistics course that used Hogg and Craig. With Volume 1's restriction to discrete sample spaces, Feller achieved considerable rigor without needing measure theory. Even in Volume 2, his use of measure theory was generally approachable. Both Volumes are still available, over 40 years since the author's death.