How do you read the summa symbol with no superscript and the Real numbers subscript? How do you read $\int_\mathbb R f(x) dx$? I'm doing continuous random variables in probability, for context, so the whole thing is:
$\int_\mathbb R f(x) dx = \int^{\infty}_{-\infty} f(x) dx = 1 $
And I get the idea that it's just saying that the total area under the curve is 1, but how do I say the first part in English? "The integral over all the real values in the domain of f?"
 A: I may read $\int_\mathbb R f(x)\;dx$ as
"The integral over R, f of x d x"
I may say "reals" instead of R.  I may omit "of x d x".
A: For a function $f:\mathbb{R}\rightarrow\mathbb{R}$, the symbol $$\int_{\mathbb{R}}f(x)\,\mathrm{d}x$$ denotes, to be precise, denotes the Lebesgue integral of $f$ over $\mathbb{R}$ with respect to the Lebesgue measure. To make this definition precise, you would need to recur to measure theory, which is actually fundamental for studying probability theory.
However, in English prose, one would read this aloud simply as the integral of $f$ over $\mathbb{R}$, omitting the fact this is the Lebesgue integral with respect to the Lebesgue measure, since this would be implied from context.
By the way, if this denotation seems strange to you, then perhaps we should revisit more familiar integral notation and try to make sense of it in that context. In more elementary mathematics, you encounter the Darboux integral, which is defined for functions $f:[a,b]\rightarrow\mathbb{R}$ rather than functions $f:\mathbb{R}\rightarrow\mathbb{R}$. We denote such an integral as $$\int_a^bf(x)\,\mathrm{d}x$$ as it is historical tradition. However, this is not the only way to denote such an integral, and although in elementary mathematics, we use this notation, mathematicians today typically tend to prefer the notation $$\int_{[a,b]}f(x)\,\mathrm{d}x$$ for a variety of reasons. For instance, when working with functions $f:U\subset\mathbb{R}^n\rightarrow\mathbb{R}$, one can generalize the definition of the Darboux integral to these functions in the way that most vector calculus students are familiar with. The "change of variables" theorem is then stated as follows: consider $g:U\subset\mathbb{R}^n\rightarrow{V\subset\mathbb{R}^n}$ for $n\in\mathbb{N},n\geq1$ being a differentiable function that is injective, and consider $f:g[U]\subset{V}\rightarrow\mathbb{R}$ being Darboux integrable. Then $$\int_Uf[g(\mathbf{x})]|det(Dg)(\mathbf{x})|\,\mathrm{d}\mathbf{x}=\int_{g[U]}f(\mathbf{x})\,\mathrm{d}\mathbf{x}.$$ This is a very important theorem that is nearly impossible to express concisely and conveniently without the ability to use this type of notation for integrals. This reveals that, in actuality, the notation you are familiar with is just an alternative convention for a special case of the more general notation.
