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I am a business student with engineering background and I am studying papers published in some journals like Management Science, Operations Research, Math of OR and they use some notations and keywords which I think are related to Pure Math.

For example, many papers in the stochastic optimization topic, define probability based on measure theory while they may use conventional probability theory. Or, they use some terms like Hilbert space, etc. Can I know if these are really necessary or are used to make papers more complicated?

If these are really necessary, I would be thankful if you can suggest me some references to improve my background in any branch of Math which can be helpful from probability to calculus.

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    $\begingroup$ Generally speaking, the purpose of rigorous mathematics is not to merely make papers more complicated. Instead, it is used to be sure that certain statements in the papers are true, by providing careful and unambiguous formulations of those statements in mathematical language, and then by providing logically rigorous derivations of those statements. If you are happy to accept the truth of such statements, i.e. if knowing why they are true is not important for your work, then feel free to ignore those papers. $\endgroup$ – Lee Mosher Feb 10 at 16:09
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There are multiple levels to operations research. (Before continuing, I want to apologize to anyone about to be scandalized by my omission of their favorite journals.)

  • For some (many?) people working in industry or government, it may be sufficient to be able to "think in systems terms", identify and classify problems ("this is a queuing problem, this other thing is more of an optimization problem"), formulate models and use appropriate tools (open-source or commercial software) to handle the computational aspects. I would say that the math background for that is algebra, some linear algebra, maybe a bit of calculus, some probability theory and enough general understanding to know which models and algorithms work in a given context and which don't (e.g., don't try to use the simplex method on a problem with quadratic equality constraints). In journal terms, think INFORMS Journal on Applied Analytics (formerly known as Interfaces).
  • Moving up in sophistication (and roping in some academics), there is a niche where you not only need to be able to formulate models, but you need to be able to design or customize algorithms for solving them. (I'll include heuristics here.) The math (and, where applicable, probability theory) requirements are a bit more rigorous at this level. I think you need a decent appreciation of convexity (what it means, what things depend on it), for example. Measure / Banach / Hilbert spaces and other weirdness are not yet on the horizon. Journals that come to mind include Operations Research, the European Journal of Operational Research, the INFORMS Journal on Optimization, Decision Sciences etc.
  • Finally, there are the folks who study problems and associated models with the intent of understanding the underlying mathematics. These are the folks (mainly but not exclusively academics) who work to show that certain cuts are facet-defining for the integer polytope of some discrete optimization model, or that solutions to some stochastic process model converge in probability to a well-known (to them, not to me) distribution as the problem size grows infinitely large. (Along with computer scientists, they may also obsess about questions relating to whether a particular problem is NP-obnoxious.) You may encounter the occasional Hilbert space (or worse) here. For this level, you need quite a bit of mathematical background. In journal terms, I would put Mathematics of Operations Research into this category.

In summary, and without having said anything too specific about mathematical prerequisites, how much math you will need will depend on where you want to fit into the OR world. FWIW, I studied Hilbert spaces in both undergrad and graduate programs, taught some form of management science for about 30 years, and probably never once mentioned them. I definitely never mentioned them in any journal articles.

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I would not say that knowing such concepts is necessary unless you wish to not only apply certain results to solve OR problems, but also to understand why/how the results work mathematically. For the most part the mathematics involved will not be too advanced; see @prubin's helpful answer for the hierarchical details.

If you are really interested in how the results are derived mathematically (provided you have time), I believe a basic understanding of fields such as measure theory, and terms such as Hilbert spaces may prove very helpful in articles in journals such as Mathematics of Operations Research as it does contain rather advanced mathematics (an example is Alvarez et al. (2005)1). For the specific area you mention (measure theory), I'd recommend Schilling's Measures, Integrals and Martingales which does a great job of introducing what a measure is, and later discussing the concepts of inner product spaces and Hilbert spaces (chapters 20 - 21).

Again, especially given your academic background, I do not think you need to know these things right away - the materials listed above are solely for interest.

P.S. People usually don't intentionally make their publications more complicated than they should be.


Reference

[1] Alvarez, F. et al. (2005). Convergence of a Hybrid Projection-Proximal Point Algorithm Coupled with Approximation Methods in Convex Optimization. Mathematics of Operations Research. 30(4):966-984.

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