Analyze a hypothetical, large-scale monolithic e-commerce codebase written in Java. Identify the 'Inventory Management' sub-domain as a candidate for extraction into a separate microservice. Outline a step-by-step migration strategy that includes: 1) Defining the anti-corruption layer, 2) Establishing the API contract, 3) Data synchronization and migration patterns to ensure zero downtime, and 4) A rollback plan in case of critical failure. Discuss the implications of distributed transactions (CAP theorem) in this specific context.