Construction of measures using random variables

Often we come across a problem to construct measures having certain properties. For instance we may want to construct one with its restriction to a class of subsets  to be a desired function. We expect such a thing to exist may be because some of previous insights in, say, the finitary aspects of subject or some physical intuitions. We find such a problem in constructing the Lebesgue measure where we ask for a measure  on a suitable large class of subsets of  \mathbb{R} giving the interval a measure equal to its length. A standard trick in measure theory called Carathéodory’s extension theorem is used to construct such a measure: We define an outer measure which then allows us to have a class of measurable sets with measure being the outer measure itself. Borel sigma algebra is contained in this class of sets. So we have the required Lebesgue measure on \mathbb{R}.

Similarly, we have a problem in probability theory about infinite coin tosses where we would like to find a probability measure on a space, the outcomes of which can be looked at as an infinite product set S=\prod_{n=1}^{\infty}\{0,1\} i.e., the each outcome is a sequence of ones and zeroes corresponding to heads or tails. Assuming each outcome is independent and that head and tails are to turn up equally likely, we expect the probability that i_1,i_2,...,i_kth outcomes turn out to be a certain known sequence to be \frac{1}{2^k}. Now we ask if  we can have a probability measure on some suitably large class of events which has these properties.

We can try to construct a measure along the same lines as we did for Lebesgue measure using the Carathéodory’s extension theorem. But because we had gone through all that trouble in defining Lebesgue measure we would like to use the Lebesgue measure on [0,1] to construct the required measure. We can do this by considering the map X:[0,1] \to S which takes \omega=\sum_{k=1}^{\infty} \frac {\omega_k}{2^k} \to (\omega_1,\omega_2,\omega_3,...), where each element in [0,1] here is written in  infinite dyadic representation. We can see that this map induces a sigma algebra on S and that indeed the pushforward of the Lebesgue measure has the required properties. Probability that k of the outcomes are a certain sequence corresponds to the length of a certain interval of [0,1] length of which is  \frac {1}{2^k}.Thus we found a probablity measure with required properties by just looking at a map from [0,1] and considering its pushforward. These maps are what probabilists call random variables. Random variable is a probabilistic concept i.e., it does not depend on the probability space.

So once we have the Lebesgue measure, most of the measures we want may be looked at the pushforwards of the Lebesgue measure without the need to go through the process of  Carathéodory’s extension once again. Also once you start studying probability spaces you find that random variables are more fundamental objects of study in probability and  that probability theory is not just about studying measures that have total measure of 1.

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