Shamsa Ishaq

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162 videos

Welcome to my channel! I am dedicated to helping students conquer their fear of mathematics. By transforming complex theories into simple, structured, and ea...

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Countably additivity of Integration

Additivity over Domain of Integration.

Monotonicity (General Integration)

Linearity part 2(General Integration)

Linearity part 1(General Integration)

The Integral Comparison Test

Properties of Lebesgue Integrable function

Definition of Lebesgue Integrable functions

Lebesgue Dominated Convergence Theorem

Monotone Convergence Theorem(part 2)

Monotone Convergence Theorem

Examples of Fatou's Lemma

Fatou's Lemma

If convergence is uniform then the limit of the integral of sequence is equal to integral of limit.

Uniform convergence of sequence of measurable functions

Pointwise a.e convergence

Limit sup of sequence of measurable function is measurable.

Infimum function of sequence of measurable function is measurable.

Supremum Function of sequence of measurable function is measurable.

Sequence of measurable function and related functions.

Sequence of measurable functions and related functions

Simple Function

Indicator function (Characteristic Function) of a set.

Positive and negative part of a function.

Minimum of finite collection of measurable functions is measurable.

Maximum of finite family of measurable function is measurable.

Composition of measurable and continuous function is measurable.

Example (Limit superior and Limit Inferior)

Limit superior and limit inferior (Another definition)

Limit inferior of a sequence of numbers