Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Build a finite sigma-algebra from observable groups.
- Justify the conclusion "The event {1} is not measurable in this observation sigma-algebra" using the stated assumptions.
Before you start
Sets and probability measures.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For Ω={1,2,3,4}, suppose observation reveals only whether the result is in A={1,2}. What events are observable?
Why this math matters
Build a finite sigma-algebra from observable groups. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The sigma-algebra describes the available observation.
- The underlying four outcomes are equally likely.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Specify which events a probability model can distinguish
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For Ω={1,2,3,4}, suppose observation reveals only whether the result is in A={1,2}. What events are observable?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
The generated sigma-algebra is {∅,A,Aᶜ,Ω}
Complements and countable unions of these sets add no finer distinctions.
Work through the mathematics
Under a uniform measure, P(A)=P(Aᶜ)=1/2
Probability assigns masses to the observable atoms.
Check the conclusion
The event {1} is not measurable in this observation sigma-algebra
The coarse model cannot distinguish one from two, even though a finer model could.
The result
The event {1} is not measurable in this observation sigma-algebra
The coarse model cannot distinguish one from two, even though a finer model could.
Common mistakes to catch
- Measurability is relative to a specified sigma-algebra.
- Unobservable does not mean probability zero in every finer model.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Is the constant function measurable?
Show a hint
Its event preimages are empty or the whole space.
Reveal answer and explanation
Yes
Those two sets always belong to a sigma-algebra.
Practice 2
Can a measurable function on this coarse space assign different values to 1 and 2?
Show a hint
Check a level set separating them.
Reveal answer and explanation
No
Such a preimage would split the atom A, which is not allowed.
Take the idea with you
Model the information lost when a sensor reports a category rather than an exact state.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Project a random variable onto partial information
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