caz said:
Yes. It could be covered in a theoretical calculus class or a real analysis class. It is more of a question of what level of sophistication your university teaches it. Check with your math department. Depending on how they present the topic, you might have to take an analysis class in order to prepare.
Thank you a lot for your answer, my first message is a bit ambiguous and I apologize for that.
Indeed I would like to self study measure theory since I have no class on that, but I have a good reference (final goal I can say) for measure theory (and integration), there is the table of content below :
I Measure theory
1 Algebras and tribes of parts of a set
1.1 Definitions
1.2 Generated tribe, reciprocal image tribe
1.3 Examples
1.4 Product of measurable spaces
1.5 The Borelian tribe
2 Measure, measured space
2.1 Definitions
2.2 Elementary properties, characterisation of a finite measure
3 Extension of a measure and applications
3.1 Extension Theorem
3.2 External measure
3.3 Application: the Borel-Lebesgue measure
3.4 Application: Stieljes Measures on R
3.5 Finite product of a family of measured spaces
4 Complements: complete tribes, regularity of measures
4.1 Negligible sets, tribe and completed measure
4.2 Regularity of measures on a metric space
5 Measurable applications
5.1 Definition of a measurable application
5.2 General properties
5.3 Properties of real measurable functions
5.4 Function with value in R¯ = [-∞, +∞]
5.5 Transport of a measure, image measure
5.6 Approximation of a real measurable function
6 Measurement theory and probability
6.1 Introduction
6.2 Elementary Examples
6.2.1 Finite set: Ω = {ω1, ...ωn}
6.2.2 Case of an infinite enumerable set Ω = {ωi, i ∈ N}
6.3 Conditional probabilities, independent events
6.4 Random Variables
6.4.1 Real Random Variables
6.4.2 Random variables, random vectors, independence
II Integration
7 Integration of Positive Measurable Functions
7.1 (Higher) Integral of Stepped Functions
7.2 Integral of a positive measurable function
7.3 Property true almost everywhere
7.4 General Properties
7.5 Transfer theorem (change of variable)
7.6 Measures defined by densities
7.7 Absolute continuous, foreign measures
7.8 Absolute continuity and density
7.9 Change of variable theorem, λ Lebesgue measure
7.10 Characterisation of the product measure, Fubini-Tonelli theorem
8 Integration of any measurable functions
8.1 Integrale of a measurable function
8.1.1 Definitions
8.1.2 The set L1
8.2 General Properties
8.2.1 First properties, Fatou's lemma
8.2.2 Theorem of domin'ee convergence and applications
8.2.3 Examples
8.3 Fubini's Theorem for any measurable functions
8.3.1 Fubini's Theorem
8.3.2 Examples
8.4 Convolution
8.4.1 Convolution of Two Measures
8.4.2 Convolution of a Function and a Measure, of Two Functions
8.4.3 Examples
9 Integration Theory and Probability
9.1 Expectation and Moments
9.1.1 Expectation
9.1.2 Moments
9.1.3 Covariance and correlation
9.1.4 Properties of moments
9.1.5 Inequalities
9.2 Real random variable (random vector) and density
9.3 Return on independence
III Functional Analysis
10 The spaces Lp and Lp, p ∈ N* ∪ +{∞}
10.1 Definitions of Lp spaces
10.1.1 The spaces Lp, p ∈ N*
10.1.2 The spaces L∞, L∞
10.2 Properties of Lp spaces, 1 ≤ p ≤ +∞
10.2.1 ||.||p is a norm
10.2.2 Completeness of Lp spaces
10.2.3 Other Properties
10.3 Dual of Lp-spaces
10.4 Some results of functional analysis in L1(R, BR, λ)
11 The Fourier transform
11.1 Definitions
11.2 General properties
11.3 Examples
11.4 General properties X = Rd: injectivity and inversion theorems
11.4.1 Injectivity Theorem
11.4.2 Inversion Theorem
11.5 Analytical properties (on R)
11.6 Fourier transform in L1: analytical properties
11.7 Fourier transform in L2
Sorry if some notations are unusual this is from my reference's lecture notes which is in french
