Mathematics: Mathematical Foundations of Probability

This course deepens your understanding of probability theory by providing a rigorous foundation for results you may have used in other courses on probability or statistics, including stochastic processes. Built on Kolmogorov’s axiomatic framework, this theory develops measure theory, including the Lebesgue integral, to establish all major results in probability. While you may encounter the Central Limit Theorem or the Law of Large Numbers in other courses, only MATM30 provides rigorous proofs. It introduces new concepts based on measure theory and proves essential results for a broad range of mathematics, such as the zero-one laws. 

Key topics include

• Sigma-algebras, Borel sigma-algebras, random variables
• Lebesgue integration and convergence theorems
• Conditional expectation with respect to a sigma-algebra
• Weak convergence in metric spaces
• Continuity theorem
• Central Limit Theorem
• Zero-One Law

The course is alternative-compulsory within the Master’s Programme in Mathematics at Lund University. It is also available as a stand-alone course, and can be taken as an elective course by Bachelor’s students in mathematics.

Teaching consists of lectures and problem-solving sessions, with a strong emphasis on active participation. You are expected to engage in discussions and work through challenging problems together with your peers.

Assessment consists of an oral examination covering both theory and applied problem-solving. If needed, a re-examination will be offered shortly after the regular exam. Adapted examination formats are available for students with permanent disabilities.

This course provides a solid foundation for advanced studies in probability, statistics, stochastic processes, and mathematical analysis. It aligns well with similar courses offered in mathematics programs worldwide.

It is particularly valuable if you aim for a research career requiring deep mathematical insight, such as academic work in mathematics, theoretical physics, data science, or financial and actuarial mathematics.

Prerequisites

English 6/English B and knowledge equivalent to 75 credits in mathematics and mathematical statistics are required.

Selection criteria

Seats are allocated according to: ECTS (HPAV): 100 %.

Tuition fees for non-EU/EEA citizens

Citizens of countries outside:

• The European Union (EU)
• The European Economic Area (EEA) and
• Switzerland

are required to pay tuition fees. You pay an instalment of the tuition fee in advance of each
semester.

Tuition fees, payments and exemptions

Full programme/course tuition fee: SEK 23,125
First payment: SEK 23,125

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Note that you may also need to pay an application fee, or provide proof of exemption.

Application fee

No tuition fees for citizens of the EU, EEA and Switzerland

There are no tuition fees for citizens of the European Union (EU), the European Economic Area (EEA) and Switzerland.