I will talk about both the role of measure theory in probability and the use pf advanced mathematics in econmics, but keep the two separate.

Probability is a field with history that goes back long before measure theory and developed to a large part independently. There is a probabilistic kernel (pun not intended) that is independent of the measure theoreic machinery. There is a beutiful book by Emmanuel Lesigne, Heads or Tails: An Introduction to Limit Theorems in Probability, that demonstrates that one can go quite far without employing measure theoretic concepts. Also, there are still competing approaches in probability to the measure theoretic approach. Edward Nelson has written a terse little book called Radically Elementary Probability Theory, in which he gives very general versions of standard probability results in terms of infinitesimals. There is also an approach to probability that is based on gambling ideas due to Glenn Shafer and Vladimier Vovk, for which they have a nice homepage.

The level of mathematical rigor in economics is not uniform. It is true that the standards of rigor in theoretical economics are essentially the same as in pure mathematics. But it is not hard to find non-rigorous arguments in top level economics journals. And it is not entirely clear that this is all bad. An example is the modelling of "ideosyncratic risk". Large populations of agents are often modelled as a continuum and if all agents face some private, independently and identically distributed risk, we can assume that risk "cancels out in the aggregate". This idea is guided by a sound intuition, but is very hard to make rigorous. The first satisfactory approach to a law of large numbers for a continuum of random variables is in a 2006 paper of Yeneng Sun. To show that an appropriate framework exists at all, Sun had to employ non-trivial machinery from non-standard analysis. The first paper that showed how to construct the appropriate spaces with standard machinery was published in 2010 (an older working paper version can be found here). I don't think it would have been very reasonable not to use the intuitive idea of individual risk cancelling out in the aggregate just because there was no way to make it completely rigorous.

The mathematics economists use is quite special. Sophisticated functional analytic concepts are used- but all vector spaces are real vector spaces. A good overview of the kind of mathematics employed in economic theory is the book Infinite Dimensional Analysis by Roko Aliprantis and Kim Border. Many fields of mathematics are completely foreign to most economic theorists. Don't expect economic theorists to know any complex analysis (econometricians may do). What is true is that a lot of mathematics in economics is qualitative, not quantitative. "An equilibrium exists. It is efficient. It is locally unique...". It is very hard to do this in a heuristic way.

I will talk about both the role of measure theory in probability and the use pf advanced mathematics in econmics, but keep the two separate.

Probability is a field with history that goes back long before measure theory and developed to a large part independently. There is a probabilistic kernel (pun not intended) that is independent of the measure theoreic machinery. There is a beutiful book by Emmanuel Lesigne, Heads or Tails: An Introduction to Limit Theorems in Probability, that demonstrates that one can go quite far without employing measure theoretic concepts. Also, there are still competing approaches in probability to the measure theoretic approach. Edward Nelson has written a terse little book called Radically Elementary Probability Theory, in which he gives very general versions of standard probability results in terms of infinitesimals. There is also an approach to probability that is based on gambling ideas due to Glenn Shafer and Vladimier Vovk, for which they have a nice homepage.

The level of mathematical rigor in economics is not uniform. It is true that the standards of rigor in theoretical economics are essentially the same as in pure mathematics. But it is not hard to find non-rigorous arguments in top level economics journals. And it is not entirely clear that this is all bad. An example is the modelling of "ideosyncratic risk". Large populations of agents are often modelled as a continuum and if all agents face some private, independently and identically distributed risk, we can assume that risk "cancels out in the aggregate". This idea is guided by a sound intuition, but is very hard to make rigorous. The first satisfactory approach to a law of large numbers for a continuum of random variables is in a 2006 paper of Yeneng Sun. To show that an appropriate framework exists at all, Sun had to employ non-trivial machinery from non-standard analysis. The first paper that showed how to construct the appropriate spaces with standard machinery was published in 2010 (an older working paper version can be found here). I don't think it would have been very reasonable not to use the intuitive idea of individual risk cancelling out in the aggregate just because there was no way to make it completely rigorous.

The mathematics economists use is quite special. Sophisticated functional analytic concepts are used- but all vector spaces are real vector spaces. A good overview of the kind of mathematics employed in economic theory is the book Infinite Dimensional Analysis by Roko Aliprantis and Kim Border. Many fields of mathematics are completely foreign to most economic theorists. Don't expect economic theorists to know any complex analysis (econometricians may do). What is true is that a lot of mathematics in economics is qualitative, not quantitative. "An equilibrium exists. It is efficient. It is locally unique...". It is very hard to do this in a heuristic way.

I will talk about both the role of measure theory in probability and the use pf advanced mathematics in econmics, but keep the two separate.

Probability is a field with history that goes back long before measure theory and developed to a large part independently. There is a probabilistic kernel (pun not intended) that is independent of the measure theoreic machinery. There is a beutiful book by Emmanuel Lesigne, Heads or Tails: An Introduction to Limit Theorems in Probability, that demonstrates that one can go quite far without employing measure theoretic concepts. Also, there are still competing approaches in probability to the measure theoretic approach. Edward Nelson has written a terse little book called Radically Elementary Probability Theory, in which he gives very general versions of standard probability results in terms of infinitesimals. There is also an approach to probability that is based on gambling ideas due to Glenn Shafer and Vladimier Vovk, for which they have a nice homepage.

The level of mathematical rigor in economics is not uniform. It is true that the standards of rigor in theoretical economics are essentially the same as in pure mathematics. But it is not hard to find non-rigorous arguments in top level economics journals. And it is not entirely clear that this is all bad. An example is the modelling of "ideosyncratic risk". Large populations of agents are often modelled as a continuum and if all agents face some private, independently and identically distributed risk, we can assume that risk "cancels out in the aggregate". This idea is guided by a sound intuition, but is very hard to make rigorous. The first satisfactory approach to a law of large numbers for a continuum of random variables is in a 2006 paper of Yeneng Sun. To show that an appropriate framework exists at all, Sun had to employ non-trivial machinery from non-standard analysis. The first paper that showed how to construct the appropriate spaces with standard machinery was published in 2010 (an older working paper version can be found here). I don't think it would have been very reasonable not to use the intuitive idea of individual risk cancelling out in the aggregate just because there was no way to make it completely rigorous.

The mathematics economists use is quite special. Sophisticated functional analytic concepts are used- but all vector spaces are real vector spaces. A good overview of the kind of mathematics employed in economic theory is the book Infinite Dimensional Analysis by Roko Aliprantis and Kim Border. Many fields of mathematics are completely foreign to most economic theorists. Don't expect economic theorists to know any complex analysis (econometricians may do). What is true is that a lot of mathematics in economics is qualitative, not quantitative. "An equilibrium exists. It is efficient. It is locally unique...". It is very hard to do this in a heuristic way.

I will talk about both the role of measure theory in probability and the use pf advanced mathematics in econmics, but keep the two separate.

Probability is a field with history that goes back long before measure theory and developed to a large part independently. There is a probabilistic kernel (pun not intended) that is independent of the measure theoreic machinery. There is a beutiful book by Emmanuel Lesigne, Heads or Tails: An Introduction to Limit Theorems in Probability, that demonstrates that one can go quite far without employing measure theoretic concepts. Also, there are still competing approaches in probability to the measure theoretic approach. Edward Nelson has written a terse little book called Radically Elementary Probability Theory, in which he gives very general versions of standard probability results in terms of infinitesimals. There is also an approach to probability that is based on gambling ideas due to Glenn Shafer and Vladimier Vovk, for which they have a nice homepage.

The level of mathematical rigor in economics is not uniform. It is true that the standards of rigor in theoretical economics are essentially the same as in pure mathematics. But it is not hard to find non-rigorous arguments in top level economics journals. And it is not entirely clear that this is all bad. An example is the modelling of "ideosyncratic risk". Large populations of agents are often modelled as a continuum and if all agents face some private, independently and identically distributed risk, we can assume that risk "cancels out in the aggregate". This idea is guided by a sound intuition, but is very hard to make rigorous. The first satisfactory approach to a law of large numbers for a continuum of random variables is in a 2006 paper of Yeneng Sun. To show that an appropriate framework exists at all, Sun had to employ non-trivial machinery from non-standard analysis. The first paper that showed how to construct the appropriate spaces with standard machinery was published in 2010 (an older working paper version can be found here). I don't think it would have been very reasonable not to use the intuitive idea of individual risk cancelling out in the aggregate just because there was no way to make it completely rigorous.

The mathematics economists use is quite special. Sophisticated functional analytic concepts are used- but all vector spaces are real vector spaces. A good overview of the kind of mathematics employed in economic theory is the book Infinite Dimensional Analysis by Roko Aliprantis and Kim Border. Many fields of mathematics are completely foreign to most economic theorists. Don't expect economic theorists to know any complex analysis (econometricians may do). What is true is that a lot of mathematics in economics is qualitative, not quantitative. "An equilibrium exists. It is efficient. It is locally unique...". It is very hard to do this in a heuristic way.