# Sigma algebra generated by random variable on a set with generators

Cross Validated Asked by Gabriel on July 28, 2020

I’m having trouble proving an intuitive result I found in these lecture notes I’m using for self-study (1.2.14 there).

Suppose $$X$$ is a $$(mathbb{S}, mathcal{S})$$-valued random variable (from $$(Omega, mathcal{F})$$), and furthermore $$mathcal{S} = sigma(mathcal{A})$$. If $$mathcal{F}^X$$ is the $$sigma$$-algebra generated by $$X$$ in $$Omega$$, we want to show that $$mathcal{F}^X = sigma({X^{-1}(A) : A in mathcal{A}})$$.

It’s easy to prove that $$mathcal{F}^X supset sigma({X^{-1}(A) : A in mathcal{A}})$$, by noticing that (i) $$mathcal{F}^X$$ is a $$sigma$$-algebra, and that (ii) it contains $${X^{-1}(A) : A in mathcal{A}}$$. But I believe I’m missing the right proof strategy for the other direction. Just appealing to the definitions and the tools developed so far (e.g. the $$pi-lambda$$ theorem) didn’t take me very far.

I think I get the spirit of the claim. Basically, it says that if you have a set of generators $$mathcal{A}$$ of $$mathcal{S}$$, to obtain $$mathcal{F}^X$$ you can either take the inverse images of all sets generated by $$mathcal{A}$$, or you can take the inverse images of just the sets in $$mathcal{A}$$ and then use those to generate a $$sigma$$-algebra. So, the order of the operations of "taking inverse images" and "generating a $$sigma$$-algebra" doesn’t matter. Is this understanding correct?

Any hint on a direction that might work for the proof would be extremely appreciated!

## Related Questions

### How does Generalized Policy Iteration stabilize to the optimal policy and value function?

1  Asked on December 6, 2021

### Fire an alert when number of sign up in an app drops. How to find the best condition to maximize accuracy?

2  Asked on December 6, 2021 by omm-kreate

### Is it always possible a closed form solution for a norm minimization problem? Which one is the best approach closed form solution or gradient based?

0  Asked on December 6, 2021 by lakshman-mahto

### Does gradient descent work for tabular Q learning?

1  Asked on December 6, 2021

### prove change in total probability of success in binomial distribution

1  Asked on December 6, 2021 by rambalachandran

### Why we cannot take baseline as predictor for change in this case

0  Asked on December 6, 2021

### Calculate group with highest defective rate

0  Asked on December 6, 2021 by user6883405

### Time series model for multiple different series observations

1  Asked on December 6, 2021

### Whitening a dataset with fewer observations than variables

1  Asked on December 5, 2021 by laos

### Composite Scores and Standardized Composite Scores t test

1  Asked on December 5, 2021 by user41710

### The distribution of the product of a multivariate normal and a lognormal distribution

1  Asked on December 5, 2021 by aae

### How to understand mapping function of kernel?

1  Asked on December 5, 2021

### Attention Mechanisms and Alignment Models in Machine Translation

1  Asked on December 5, 2021

### Difference between Repeated measures ANOVA, ANCOVA and Linear mixed effects model

1  Asked on December 5, 2021

### Time Series Multivariate Forecasting

1  Asked on December 5, 2021

### reporting results of a multivariate logistic regression using the glm function in R

1  Asked on December 5, 2021 by b-kenobi

### Checking the constant variance assumption for residuals vs fitted plots: What about for the same fitted values?

1  Asked on December 5, 2021

### What model is a suitable model for zero-constrained variables?

0  Asked on December 5, 2021

### Why the regression coefficient for normalized continuous variable is unexpected when there is dummy variable in the model?

1  Asked on December 5, 2021 by emberbillow

### Is boosting and bagging only relevant in the context of decision trees?

2  Asked on December 5, 2021

### Ask a Question

Get help from others!