I'm currently learning about stochastic calculus for the first time. I have taken a first course in Real Analysis covering Lebesgue integration.
According to Wikipedia, we write out the Itô stochastic integral as
$$Y_t = \int_0^{t}H_s \,dX_s$$
where "$H$ is a locally square-integrable process adapted to the filtration generated by $X$, which is a Brownian motion or, more generally, a semimartingale." I'm confused because $X_s$ is in the same location in our integral as a measure would be in Lebesgue integration. According to Wikipedia, we write the Lebesgue integral as
$$\int f \,d\mu$$
where $\mu$ is a measure on the domain of $f$ and $f$ is measurable with respect to $\mu$.
In finance, we often use $X_s$ to model the price of a security. $X_s$, or changes in $X_s$, can be negative. This makes it difficult to connect semimartingales and measures.
How is $X_s$ related to measures? Are there ways to interpret $X_s$ as a measure?
As noted in the comments, the concept of a stochastic integral is much more resembling the Lebesgue-Stieljes integral
$$\int _{a}^{b}f(x)\,dg(x)$$ which is defined for ${\displaystyle f:\left[a,b\right]\rightarrow \mathbb {R} }$ Borel-measurable and bounded and ${\displaystyle g:\left[a,b\right]\rightarrow \mathbb {R} }$ having bounded variation in $[a, b]$. I want to emphasize that the condition on $g$ to have bounded variation is crucial here, otherwise this integral is ill-defined. As it turns out, this condition is often not satisfied in the context of stochastic processes.
Rather than considering a general semimartingale $(X_t)_{t \geq 0}$, let's take a much more concrete object to make this point clear. Let $B:=(B_t)_{t \geq 0}$ be the one-dimensional standard Brownian motion, for example. It is well-known and not so hard to show that $B$ indeed has infinite variation. Hence, in order to make sense of an expression like $$ \int_{0}^{t} d B_s\ $$ which intuitively tries to capture the idea of integrating with respect to the Brownian motion $B$, we need to provide some additional justification as this transcends the assumptions we impose on $g$ in the Lebesgue-Stieljes-integral above.
Rather than explaining the construction of a stochastic integral from stratch, I refer to LeGall's succinct and precise exposition in "Brownian Motion, Martingales, and Stochastic Calculus" Ch.4 and Ch.5. In particular, 4.1.1. will help you to gain an understanding of why, as you phrase, the $X_s$ appear at the same place as $\mu$.
Finally, to address your concern that $X$ may be negative, but $\mu$ not, I encourage you to familiarize with the notion of signed measures. This is a generalization of the concept of measure by allowing it to have negative values.